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Mathematics · Class 12 Science

Ch 9Differential Equations — Class 12 Mathematics, concept-first.

In earlier classes, given you learned to find its derivative . In integral calculus you reversed this: given , find such that . This reverse problem can be written as

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Order Of Differential Equation

A differential equation involves an unknown function together with its derivatives . The order of the equation is simply the order of the highest derivative that appears in it.

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9.1

Introduction

In earlier classes, given you learned to find its derivative . In integral calculus you reversed this: given , find such that . This reverse problem can be written as

9.2

Basic Concepts

You already know equations like and , which involve only the variables themselves. Now consider

9.2.1

Order of a Differential Equation

The order of a differential equation is the order of the highest-order derivative of the dependent variable that appears in the equation. This is the first and most fundamental classification.

9.2.2

Degree of a Differential Equation

13 Q

Before we define degree, we must understand a crucial restriction. The degree of a differential equation is only defined when the equation is a polynomial equation in its derivatives.

+Worked Examplesi1 question
  1. Example 1Find the order and degree, if defined, of each of the following differential equations: (i) $\frac{dy}{dx} - \cos x = 0$ (ii) $xy\frac{d^2y}…Preview
+Exercise 9.1i12 questions
  1. Q1Determine the order and degree, if defined, of the differential equation: $\frac{d^4y}{dx^4} + \sin(y''') = 0$Free
  2. Q2Determine the order and degree, if defined, of the differential equation: $y' + 5y = 0$Free
  3. Q3Determine the order and degree, if defined, of the differential equation: $\left(\frac{ds}{dt}\right)^4 + 3s \frac{d^2s}{dt^2} = 0$Free
  4. Q4Determine the order and degree, if defined, of the differential equation: $\left(\frac{d^2y}{dx^2}\right)^2 + \cos \left(\frac{dy}{dx}\right…Preview
  5. Q5Determine the order and degree, if defined, of the differential equation: $\frac{d^2y}{dx^2} = \cos 3x + \sin 3x$Preview
  6. Q6Determine the order and degree, if defined, of the differential equation: $(y''')^2 + (y'')^3 + (y')^4 + y^5 = 0$Preview
  7. Q7Determine the order and degree, if defined, of the differential equation: $y''' + 2y'' + y' = 0$Preview
  8. Q8Determine the order and degree, if defined, of the differential equation: $y' + y = e^x$Preview
  9. Q9Determine the order and degree, if defined, of the differential equation: $y'' + (y')^2 + 2y = 0$Preview
  10. Q10Determine the order and degree, if defined, of the differential equation: $y'' + 2y' + \sin y = 0$Preview
  11. Q11Determine the order and degree, if defined, of the differential equation: The degree of the differential equation $\left(\frac{d^2 y}{dx^2}\…Preview
  12. Q12The order of the differential equation $2x^2 \dfrac{d^2 y}{dx^2} - 3 \dfrac{dy}{dx} + y = 0$ is (A) 2 (B) 1 (C) 0 (D) not definedPreview
9.3

General and Particular Solutions of a Differential Equation

14 Q

In earlier classes, you solved equations like or . The solution to such an equation is a number (real or complex) that, when substituted for the unknown , makes the left-hand side equal to the right-h…

+Worked Examplesi2 questions
  1. Example 2Verify that the function $y = e^{-3x}$ is a solution of the differential equation $\frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 0$.Free
  2. Example 3Verify that the function $y = a\cos x + b\sin x$, where $a, b \in \mathbf{R}$ is a solution of the differential equation $\frac{d^2y}{dx^2}…Preview
+Exercise 9.2i12 questions
  1. Q1Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y = e^x + 1$ : $y'' - y' =…Free
  2. Q2Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y = x^2 + 2x + C$ : $y' - 2…Free
  3. Q3Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y = \cos x + C$ : $y' + \si…Free
  4. Q4Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y = \sqrt{1 + x^2}$ : $y' =…Preview
  5. Q5Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $y = Ax$ : $xy' = y \ (x \ne…Preview
  6. Q6Solve the following differential equation: $y = x \sin x$ ; $xy' = y + x \sqrt{x^2 - y^2}$ ($x \neq 0$ and $x > y$ or $x < -y$)Preview
  7. Q7Verify that the given implicit function is a solution of the corresponding differential equation: $xy = \log y + C$ ; $y' = \frac{y^2}{1-xy}…Preview
  8. Q8Solve the following differential equation: $y - \cos y = x$ ; $(y \sin y + \cos y + x) y' = y$Preview
  9. Q9Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation: $x + y = \tan^{-1} y$ : $y^2…Preview
  10. Q10Solve the following differential equation: $y = \sqrt{a^2 - x^2}$ $x \in (-a, a)$ ; $x + y \frac{dy}{dx} = 0 (y \neq 0)$Preview
  11. Q11The number of arbitrary constants in the general solution of a differential equation of fourth order are: (A) 0 (B) 2 (C) 3 (D) 4Preview
  12. Q12The number of arbitrary constants in the particular solution of a differential equation of third order are: (A) 3 (B) 2 (C) 1 (D) 0Preview
9.4

Methods of Solving First Order, First Degree Differential Equations

A differential equation of the form is called first order, first degree: the highest derivative present is the first, and it appears only to the first power.

9.4.1

Differential Equations with Variables Separable

29 Q

A first-order, first-degree differential equation has the general form

+Worked Examplesi6 questions
  1. Example 4Find the general solution of the differential equation $\frac{dy}{dx} = \frac{1+x}{2-y}$, $(y \neq 2)$.Free
  2. Example 5Find the general solution of the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$.Free
  3. Example 6Find the particular solution of the differential equation $\frac{dy}{dx} = -4xy^2$ given that $y = 1$, when $x = 0$.Preview
  4. Example 7Find the equation of the curve passing through the point $(1, 1)$ whose differential equation is $x\,dy = (2x^2 + 1)\,dx$ $(x \neq 0)$.Preview
  5. Example 8Find the equation of a curve passing through the point $(-2, 3)$, given that the slope of the tangent to the curve at any point $(x, y)$ is…Preview
  6. Example 9In a bank, principal increases continuously at the rate of 5% per year. In how many years Rs 1000 double itself?Preview
+Exercise 9.3i23 questions
  1. Q1Solve the following differential equation: $\frac{dy}{dx} = \frac{1 - \cos x}{1 + \cos x}$Free
  2. Q2Solve the following differential equation: $\frac{dy}{dx} = \sqrt{4 - y^2}$ ($-2 < y < 2$)Free
  3. Q3Solve the following differential equation: $\frac{dy}{dx} + y = 1 (y \neq 1)$Free
  4. Q4Solve the following differential equation: $\sec^2 x \tan y \, dx + \sec^2 y \tan x \, dy = 0$Preview
  5. Q5Solve the following differential equation: $(e^x + e^{-x}) \, dy - (e^x - e^{-x}) \, dx = 0$Preview
  6. Q6Solve the following differential equation: $\frac{dy}{dx} = (1+x^2)(1+y^2)$Preview
  7. Q7Solve the following differential equation: $y \log y \, dx - x \, dy = 0$Preview
  8. Q8Solve the following differential equation: $x^5 \frac{dy}{dx} = -y^5$Preview
  9. Q9Solve the differential equation $\dfrac{dy}{dx} = \sin^{-1} x$.Preview
  10. Q10$e^x \tan y \, dx + (1 - e^x) \sec^2 y \, dy = 0$Preview
  11. Q11Solve the following differential equation: $(x^3 + x^2 + x + 1) \frac{dy}{dx} = 2x^2 + x; y = 1$ when $x = 0$Preview
  12. Q12Solve the following differential equation: $x(x^2 - 1) \frac{dy}{dx} = 1; y = 0$ when $x = 2$Preview
  13. Q13Solve the following differential equation: $\cos \left(\frac{dy}{dx}\right) = a \ (a \in \mathbf{R}); y = 1$ when $x = 0$Preview
  14. Q14Solve the following differential equation: $\frac{dy}{dx} = y \tan x; y = 1$ when $x = 0$Preview
  15. Q15Find the equation of a curve passing through the point $(0, 0)$ and whose differential equation is $y' = e^x \sin x$.Preview
  16. Q16For the differential equation $xy \frac{dy}{dx} = (x+2)(y+2)$, find the solution curve passing through the point $(1, -1)$.Preview
  17. Q17Find the equation of a curve passing through the point $(0, -2)$ given that at any point $(x, y)$ on the curve, the product of the slope of…Preview
  18. Q18At any point $(x, y)$ of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point…Preview
  19. Q19The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 un…Preview
  20. Q20In a bank, principal increases continuously at the rate of $r\%$ per year. Find the value of $r$ if Rs 100 double itself in 10 years $(\log_…Preview
  21. Q21Solve the following differential equation: In a bank, principal increases continuously at the rate of $5\%$ per year. An amount of Rs 1000 i…Preview
  22. Q22In a culture, the bacteria count is 1,00,000. The number is increased by $10\%$ in 2 hours. In how many hours will the count reach 2,00,000,…Preview
  23. Q23The general solution of the differential equation $\dfrac{dy}{dx} = e^{x+y}$ is (A) $e^x + e^{-y} = C$ (B) $e^x + e^y = C$ (C) $e^{-x} + e^y…Preview
9.4.2

Homogeneous Differential Equations

21 Q

A function is homogeneous of degree if replacing by (for any nonzero ) gives

+Worked Examplesi4 questions
  1. Example 10Show that the differential equation $(x - y)\frac{dy}{dx} = x + 2y$ is homogeneous and solve it.Free
  2. Example 11Show that the differential equation $x\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$ is homogeneous and solv…Free
  3. Example 12Show that the differential equation $2y\,e^{x/y}\,dx + \left(y - 2x\,e^{x/y}\right)dy = 0$ is homogeneous and find its particular solution,…Preview
  4. Example 13Show that the family of curves for which the slope of the tangent at any point $(x, y)$ on it is $\frac{x^2 + y^2}{2xy}$, is given by $x^2 -…Preview
+Exercise 9.4i17 questions
  1. Q1Solve the following differential equation: $(x^2 + xy) dy = (x^2 + y^2) dx$Free
  2. Q2Solve the following differential equation: $y' = \frac{x+y}{x}$Free
  3. Q3Solve the following differential equation: $(x - y) dy - (x + y) dx = 0$Free
  4. Q4Solve the following differential equation: $(x^2 - y^2) dx + 2xy dy = 0$Preview
  5. Q5Solve the following differential equation: $x^2 \frac{dy}{dx} = x^2 - 2y^2 + xy$Preview
  6. Q6Solve the following differential equation: $x dy - y dx = \sqrt{x^2 + y^2} dx$Preview
  7. Q7Solve the following differential equation: $\left\{x \cos\left(\dfrac{y}{x}\right) + y \sin\left(\dfrac{y}{x}\right)\right\} y\,dx = \left\{…Preview
  8. Q8Solve the following differential equation: $x \frac{dy}{dx} - y + x \sin \left(\frac{y}{x}\right) = 0$Preview
  9. Q9Solve the following differential equation: $y dx + x \log \left(\frac{y}{x}\right) dy - 2x dy = 0$Preview
  10. Q10$\left(1+e^{\frac{x}{y}}\right) dx + e^{\frac{x}{y}} \left(1-\frac{x}{y}\right) dy = 0$Preview
  11. Q11Solve the following differential equation: $(x + y) dy + (x - y) dx = 0; y = 1$ when $x = 1$Preview
  12. Q12Solve the following differential equation: $x^2 dy + (xy + y^2) dx = 0; y = 1$ when $x = 1$Preview
  13. Q13Solve the following differential equation: $\left[x \sin^2 \left(\frac{y}{x}\right) - y\right] dx + x dy = 0; y = \frac{\pi}{4}$ when $x = 1…Preview
  14. Q14Solve the following differential equation: $\frac{dy}{dx} - \frac{y}{x} + \text{cosec}\left(\frac{y}{x}\right) = 0; y = 0$ when $x = 1$Preview
  15. Q15Solve the following differential equation: $2xy + y^2 - 2x^2 \frac{dy}{dx} = 0; y = 2$ when $x = 1$Preview
  16. Q16A homogeneous differential equation of the form $\dfrac{dx}{dy} = h\left(\dfrac{x}{y}\right)$ can be solved by making the substitution (A) $…Preview
  17. Q17Which of the following is a homogeneous differential equation? (A) $(4x + 6y + 5)\, dy - (3y + 2x + 4)\, dx = 0$ (B) $(xy)\, dx - (x^3 + y^3…Preview
9.4.3

Linear Differential Equations

24 Q

A differential equation is linear when the dependent variable and all its derivatives appear only in the first power and are not multiplied together.

+Worked Examplesi5 questions
  1. Example 14Find the general solution of the differential equation $\frac{dy}{dx} - y = \cos x$.Free
  2. Example 15Find the general solution of the differential equation $x\frac{dy}{dx} + 2y = x^2$ $(x \neq 0)$.Free
  3. Example 16Find the general solution of the differential equation $y\,dx - (x + 2y^2)\,dy = 0$.Preview
  4. Example 17Find the particular solution of the differential equation $\frac{dy}{dx} + y\cot x = 2x + x^2 \cot x$ $(x \neq 0)$ given that $y = 0$ when $…Preview
  5. Example 18Find the equation of a curve passing through the point $(0, 1)$. If the slope of the tangent to the curve at any point $(x, y)$ is equal to…Preview
+Exercise 9.5i19 questions
  1. Q1Solve the following differential equation: $\frac{dy}{dx} + 2y = \sin x$Free
  2. Q2Solve the following differential equation: $\frac{dy}{dx} + 3y = e^{-2x}$Free
  3. Q3Solve the following differential equation: $\frac{dy}{dx} + \frac{y}{x} = x^2$Free
  4. Q4Solve the following differential equation: $\frac{dy}{dx} + (\sec x) y = \tan x \quad \left(0 \le x < \frac{\pi}{2}\right)$Preview
  5. Q5Solve the following differential equation: $\cos^2 x \frac{dy}{dx} + y = \tan x \quad \left(0 \le x < \frac{\pi}{2}\right)$Preview
  6. Q6Solve the following differential equation: $x \frac{dy}{dx} + 2y = x^2 \log x$Preview
  7. Q7Solve the following differential equation: $x \log x \frac{dy}{dx} + y = \frac{2}{x} \log x$Preview
  8. Q8Solve the following differential equation: $(1 + x^2) dy + 2xy \ dx = \cot x \ dx \quad (x \ne 0)$Preview
  9. Q9Solve the following differential equation: $x \frac{dy}{dx} + y - x + xy \cot x = 0 \quad (x \neq 0)$Preview
  10. Q10Solve the following differential equation: $(x+y)\frac{dy}{dx}=1$Preview
  11. Q11Solve the following differential equation: $y \ dx + (x - y^2) \ dy = 0$Preview
  12. Q12$(x+3y^2)\frac{dy}{dx}=y \quad (y>0)$Preview
  13. Q13Solve the following differential equation: $\frac{dy}{dx} + 2y \tan x = \sin x; \ y=0 \text{ when } x=\frac{\pi}{3}$Preview
  14. Q14Solve the following differential equation: $(1+x^2)\frac{dy}{dx} + 2xy = \frac{1}{1+x^2}; \ y=0 \text{ when } x=1$Preview
  15. Q15Solve the following differential equation: $\frac{dy}{dx} - 3y \cot x = \sin 2x; \ y=2 \text{ when } x=\frac{\pi}{2}$Preview
  16. Q16Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point $(x, y)$ is equal to t…Preview
  17. Q17Find the equation of a curve passing through the point $(0, 2)$ given that the sum of the coordinates of any point on the curve exceeds the…Preview
  18. Q18Solve the following differential equation: The Integrating Factor of the differential equation $x \frac{dy}{dx} - y = 2x^2$ is (A) $e^{-x}$…Preview
  19. Q19The Integrating Factor of the differential equation $(1 - y^2)\dfrac{dx}{dy} + yx = ay \ (-1 < y < 1)$ is (A) $\dfrac{1}{y^2 - 1}$ (B) $\dfr…Preview

Miscellaneous Examples

+Miscellaneous Examplesi4 questions
  1. Example 19Verify that the function $y = c_1 e^{ax}\cos bx + c_2 e^{ax}\sin bx$, where $c_1, c_2$ are arbitrary constants is a solution of the differen…Free
  2. Example 20Find the particular solution of the differential equation $\log\left(\frac{dy}{dx}\right) = 3x + 4y$ given that $y = 0$ when $x = 0$.Free
  3. Example 21Solve the differential equation $(x\,dy - y\,dx)\,y\sin\left(\frac{y}{x}\right) = (y\,dx + x\,dy)\,x\cos\left(\frac{y}{x}\right)$.Preview
  4. Example 22Solve the differential equation $(\tan^{-1}y - x)\,dy = (1 + y^2)\,dx$.Preview

Miscellaneous Exercise on Chapter 9

+Miscellaneous Exercisei15 questions
  1. Q1For each of the differential equations given below, indicate its order and degree (if defined). (i) $\dfrac{d^2 y}{dx^2} + 5x \left(\dfrac{d…Free
  2. Q2For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential…Free
  3. Q3Prove that $x^2 - y^2 = c(x^2 + y^2)^2$ is the general solution of differential equation $(x^3 - 3xy^2)\, dx = (y^3 - 3x^2 y)\, dy$, where $…Free
  4. Q4Find the general solution of the differential equation $\frac{dy}{dx} + \sqrt{\frac{1 - y^2}{1 - x^2}} = 0$.Preview
  5. Q5Show that the general solution of the differential equation $\frac{dy}{dx} + \frac{y^2 + y + 1}{x^2 + x + 1} = 0$ is given by $(x + y + 1) =…Preview
  6. Q6Find the equation of the curve passing through the point $\left(0, \frac{\pi}{4}\right)$ whose differential equation is $\sin x \cos y\, dx…Preview
  7. Q7Find the particular solution of the differential equation $(1 + e^{2x})\, dy + (1 + y^2) e^x\, dx = 0$, given that $y = 1$ when $x = 0$.Preview
  8. Q8Solve the differential equation $y e^{x/y}\, dx = \left(x e^{x/y} + y^2\right) dy\ (y \neq 0)$.Preview
  9. Q9Find a particular solution of the differential equation $(x - y)(dx + dy) = dx - dy$, given that $y = -1$, when $x = 0$. (Hint: put $x - y =…Preview
  10. Q10Solve the differential equation $\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}\right) \frac{dx}{dy} = 1\ (x \neq 0)$.Preview
  11. Q11Find a particular solution of the differential equation $\frac{dy}{dx} + y \cot x = 4x \operatorname{cosec} x\ (x \neq 0)$, given that $y =…Preview
  12. Q12Find a particular solution of the differential equation $(x + 1) \frac{dy}{dx} = 2 e^{-y} - 1$, given that $y = 0$ when $x = 0$.Preview
  13. Q13The general solution of the differential equation $\frac{y\, dx - x\, dy}{y} = 0$ is (A) $xy = C$ (B) $x = Cy^2$ (C) $y = Cx$ (D) $y = Cx^2$Preview
  14. Q14The general solution of a differential equation of the type $\frac{dx}{dy} + P_1 x = Q_1$ is (A) $y e^{\int P_1\, dy} = \int \left(Q_1 e^{\i…Preview
  15. Q15The general solution of the differential equation $e^x\, dy + (y e^x + 2x)\, dx = 0$ is (A) $x e^y + x^2 = C$ (B) $x e^y + y^2 = C$ (C) $y e…Preview

Summary

- Order and Degree: Order is the highest derivative present; degree is the power of the highest derivative after removing radicals and fractions.

NCERT Exemplar

Higher-order thinking problems from the NCERT Exemplar.

+Show 97 questions97 questions
  1. Q1Find the solution of $\frac{dy}{dx}=2^{y-x}$.Free
  2. Q2Find the differential equation of all non-vertical lines in a plane.Free
  3. Q3Given that $\frac{dy}{dx}=e^{-2y}$ and $y=0$ when $x=5$. Find the value of $x$ when $y=3$.Free
  4. Q4Solve the differential equation $(x^2-1)\frac{dy}{dx}+2xy=\frac{1}{x^2-1}$.Preview
  5. Q5Solve the differential equation $\frac{dy}{dx}+2xy=y$.Preview
  6. Q6Find the general solution of $\frac{dy}{dx}+ay=e^{mx}$.Preview
  7. Q7Solve the differential equation $\frac{dy}{dx}+1=e^{x+y}$.Preview
  8. Q8Solve: $y\,dx-x\,dy=x^2 y\,dx$.Preview
  9. Q9Solve the differential equation $\frac{dy}{dx}=1+x+y^2+xy^2$, when $y=0$, $x=0$.Preview
  10. Q10Find the general solution of $(x+2y^3)\frac{dy}{dx}=y$.Preview
  11. Q11If $y(x)$ is a solution of $\left(\frac{2+\sin x}{1+y}\right)\frac{dy}{dx}=-\cos x$ and $y(0)=1$, then find the value of $y\left(\frac{\pi}{…Preview
  12. Q12If $y(t)$ is a solution of $(1+t)\frac{dy}{dt}-ty=1$ and $y(0)=-1$, then show that $y(1)=-\frac{1}{2}$.Preview
  13. Q13Form the differential equation having $y=(\sin^{-1}x)^2+A\cos^{-1}x+B$, where $A$ and $B$ are arbitrary constants, as its general solution.Preview
  14. Q14Form the differential equation of all circles which pass through origin and whose centres lie on $y$-axis.Preview
  15. Q15Find the equation of a curve passing through origin and satisfying the differential equation $(1+x^2)\frac{dy}{dx}+2xy=4x^2$.Preview
  16. Q16Solve: $x^2\frac{dy}{dx}=x^2+xy+y^2$.Preview
  17. Q17Find the general solution of the differential equation $(1+y^2)+(x-e^{\tan^{-1}y})\frac{dy}{dx}=0$.Preview
  18. Q18Find the general solution of $y^2\,dx+(x^2-xy+y^2)\,dy=0$.Preview
  19. Q19Solve: $(x+y)(dx-dy)=dx+dy$. [Hint: Substitute $x+y=z$ after separating $dx$ and $dy$.]Preview
  20. Q20Solve: $2(y+3)-xy\frac{dy}{dx}=0$, given that $y(1)=-2$.Preview
  21. Q21Solve the differential equation $dy=\cos x\,(2-y\csc x)\,dx$ given that $y=2$ when $x=\frac{\pi}{2}$.Preview
  22. Q22Form the differential equation by eliminating $A$ and $B$ in $Ax^2+By^2=1$.Preview
  23. Q23Solve the differential equation $(1+y^2)\tan^{-1}x\,dx+2y(1+x^2)\,dy=0$.Preview
  24. Q24Find the differential equation of system of concentric circles with centre $(1,\,2)$.Preview
  25. Q25Solve: $y+\frac{d}{dx}(xy)=x(\sin x+\log x)$.Preview
  26. Q26Find the general solution of $(1+\tan y)(dx-dy)+2x\,dy=0$.Preview
  27. Q27Solve: $\frac{dy}{dx}=\cos(x+y)+\sin(x+y)$. [Hint: Substitute $x+y=z$.]Preview
  28. Q28Find the general solution of $\frac{dy}{dx}-3y=\sin 2x$.Preview
  29. Q29Find the equation of a curve passing through $(2,\,1)$ if the slope of the tangent to the curve at any point $(x,\,y)$ is $\frac{x^2+y^2}{2x…Preview
  30. Q30Find the equation of the curve through the point $(1,\,0)$ if the slope of the tangent to the curve at any point $(x,\,y)$ is $\frac{y-1}{x^…Preview
  31. Q31Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point $(x,\,y)$ is equal to the square o…Preview
  32. Q32Find the equation of a curve passing through the point $(1,\,1)$, if the tangent drawn at any point $P(x,\,y)$ on the curve meets the co-ord…Preview
  33. Q33Solve: $x\frac{dy}{dx}=y(\log y-\log x+1)$.Preview
  34. Q34(i) The degree of the differential equation $\frac{d^2y}{dx^2}+e^{\frac{dy}{dx}}=0$ is ______.Preview
  35. Q35(ii) The degree of the differential equation $\sqrt{1+\left(\frac{dy}{dx}\right)^2}=x$ is ______.Preview
  36. Q36(iii) The number of arbitrary constants in the general solution of a differential equation of order three is ______.Preview
  37. Q37(iv) $\frac{dy}{dx}+\frac{y}{x\log x}=\frac{1}{x}$ is an equation of the type ______.Preview
  38. Q38(v) General solution of the differential equation of the type $\frac{dx}{dy}+P_1 x=Q_1$ is given by ______.Preview
  39. Q39(vi) The solution of the differential equation $x\frac{dy}{dx}+2y=x^2$ is ______.Preview
  40. Q40(vii) The solution of $(1+x^2)\frac{dy}{dx}+2xy-4x^2=0$ is ______.Preview
  41. Q41(viii) The solution of the differential equation $y\,dx+(x+xy)\,dy=0$ is ______.Preview
  42. Q42(ix) General solution of $\frac{dy}{dx}+y=\sin x$ is ______.Preview
  43. Q43(x) The solution of the differential equation $\cot y\,dx=x\,dy$ is ______.Preview
  44. Q44(xi) The integrating factor of $\frac{dy}{dx}+y=\frac{1+y}{x}$ is ______.Preview
  45. Q45(i) Integrating factor of the differential equation of the form $\frac{dx}{dy}+p_1 x=Q_1$ is given by $e^{\int p_1\,dy}$. (State True or Fal…Preview
  46. Q46(ii) Solution of the differential equation of the type $\frac{dx}{dy}+p_1 x=Q_1$ is given by $x\cdot(\text{I.F.})=\int (\text{I.F.})\times Q…Preview
  47. Q47(iii) Correct substitution for the solution of the differential equation of the type $\frac{dy}{dx}=f(x,y)$, where $f(x,y)$ is a homogeneous…Preview
  48. Q48(iv) Correct substitution for the solution of the differential equation of the type $\frac{dx}{dy}=g(x,y)$, where $g(x,y)$ is a homogeneous…Preview
  49. Q49(v) Number of arbitrary constants in the particular solution of a differential equation of order two is two. (State True or False.)Preview
  50. Q50(vi) The differential equation representing the family of circles $x^2+(y-a)^2=a^2$ will be of order two. (State True or False.)Preview
  51. Q51(vii) The solution of $\frac{dy}{dx}=\left(\frac{y}{x}\right)^{1/3}$ is $y^{2/3}-x^{2/3}=c$. (State True or False.)Preview
  52. Q52(viii) Differential equation representing the family of curves $y=e^x(A\cos x+B\sin x)$ is $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0$. (State T…Preview
  53. Q53(ix) The solution of the differential equation $\frac{dy}{dx}=\frac{x+2y}{x}$ is $x+y=kx^2$. (State True or False.)Preview
  54. Q54(x) Solution of $\frac{dy}{dx}=\frac{y}{x}+\tan\frac{y}{x}$ is $\sin\left(\frac{y}{x}\right)=cx$. (State True or False.)Preview
  55. Q55(xi) The differential equation of all non-horizontal lines in a plane is $\frac{d^2x}{dy^2}=0$. (State True or False.)Preview
  56. Q56The degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^2+\left(\frac{dy}{dx}\right)^2=x\sin\left(\frac{dy}{dx}\right)$ is:…Preview
  57. Q57The degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]^{3/2}=\frac{d^2y}{dx^2}$ is: (A) 4 (B) $\frac{3}{2}$ (C…Preview
  58. Q58The order and degree of the differential equation $\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^{1/4}+x^{1/5}=0$, respectively, are: (A) 2 a…Preview
  59. Q59If $y=e^{-x}(A\cos x+B\sin x)$, then $y$ is a solution of: (A) $\frac{d^2y}{dx^2}+2\frac{dy}{dx}=0$ (B) $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y…Preview
  60. Q60The differential equation for $y=A\cos\alpha x+B\sin\alpha x$, where $A$ and $B$ are arbitrary constants, is: (A) $\frac{d^2y}{dx^2}-\alpha^…Preview
  61. Q61Solution of differential equation $x\,dy-y\,dx=0$ represents: (A) a rectangular hyperbola (B) parabola whose vertex is at origin (C) straigh…Preview
  62. Q62Integrating factor of the differential equation $\cos x\frac{dy}{dx}+y\sin x=1$ is: (A) $\cos x$ (B) $\tan x$ (C) $\sec x$ (D) $\sin x$Preview
  63. Q63Solution of the differential equation $\tan y\,\sec^2 x\,dx+\tan x\,\sec^2 y\,dy=0$ is: (A) $\tan x+\tan y=k$ (B) $\tan x-\tan y=k$ (C) $\fr…Preview
  64. Q64Family $y=Ax+A^3$ of curves is represented by the differential equation of degree: (A) 1 (B) 2 (C) 3 (D) 4Preview
  65. Q65Integrating factor of $x\frac{dy}{dx}-y=x^4-3x$ is: (A) $x$ (B) $\log x$ (C) $\frac{1}{x}$ (D) $-x$Preview
  66. Q66Solution of $\frac{dy}{dx}-y=1$, $y(0)=1$ is given by: (A) $xy=-e^x$ (B) $xy=-e^{-x}$ (C) $xy=-1$ (D) $y=2e^x-1$Preview
  67. Q67The number of solutions of $\frac{dy}{dx}=\frac{y+1}{x-1}$ when $y(1)=2$ is: (A) none (B) one (C) two (D) infinitePreview
  68. Q68Which of the following is a second order differential equation? (A) $(y')^2+x=y^2$ (B) $y'y''+y=\sin x$ (C) $y'''+(y'')^2+y=0$ (D) $y'=y^2$Preview
  69. Q69Integrating factor of the differential equation $(1-x^2)\frac{dy}{dx}-xy=1$ is: (A) $-x$ (B) $\frac{x}{1+x^2}$ (C) $\sqrt{1-x^2}$ (D) $\frac…Preview
  70. Q70$\tan^{-1}x+\tan^{-1}y=c$ is the general solution of the differential equation: (A) $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$ (B) $\frac{dy}{dx}=\…Preview
  71. Q71The differential equation $y\frac{dy}{dx}+x=c$ represents: (A) Family of hyperbolas (B) Family of parabolas (C) Family of ellipses (D) Famil…Preview
  72. Q72The general solution of $e^x\cos y\,dx-e^x\sin y\,dy=0$ is: (A) $e^x\cos y=k$ (B) $e^x\sin y=k$ (C) $e^x=k\cos y$ (D) $e^x=k\sin y$Preview
  73. Q73The degree of the differential equation $\frac{d^2y}{dx^2}+\left(\frac{dy}{dx}\right)^3+6y^5=0$ is: (A) 1 (B) 2 (C) 3 (D) 5Preview
  74. Q74The solution of $\frac{dy}{dx}+y=e^{-x}$, $y(0)=0$ is: (A) $y=e^x(x-1)$ (B) $y=xe^{-x}$ (C) $y=xe^{-x}+1$ (D) $y=(x+1)e^{-x}$Preview
  75. Q75Integrating factor of the differential equation $\frac{dy}{dx}+y\tan x-\sec x=0$ is: (A) $\cos x$ (B) $\sec x$ (C) $e^{\cos x}$ (D) $e^{\sec…Preview
  76. Q76The solution of the differential equation $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$ is: (A) $y=\tan^{-1}x$ (B) $y-x=k(1+xy)$ (C) $x=\tan^{-1}y$ (D…Preview
  77. Q77The integrating factor of the differential equation $\frac{dy}{dx}+y=\frac{1+y}{x}$ is: (A) $\frac{x}{e^x}$ (B) $\frac{e^x}{x}$ (C) $xe^x$ (…Preview
  78. Q78$y=ae^{mx}+be^{-mx}$ satisfies which of the following differential equation? (A) $\frac{dy}{dx}+my=0$ (B) $\frac{dy}{dx}-my=0$ (C) $\frac{d^…Preview
  79. Q79The solution of the differential equation $\cos x\sin y\,dx+\sin x\cos y\,dy=0$ is: (A) $\frac{\sin x}{\sin y}=c$ (B) $\sin x\sin y=c$ (C) $…Preview
  80. Q80The solution of $x\frac{dy}{dx}+y=e^x$ is: (A) $y=\frac{e^x}{x}+\frac{k}{x}$ (B) $y=xe^x+cx$ (C) $y=xe^x+k$ (D) $x=\frac{e^y}{y}+\frac{k}{y}…Preview
  81. Q81The differential equation of the family of curves $x^2+y^2-2ay=0$, where $a$ is arbitrary constant, is: (A) $(x^2-y^2)\frac{dy}{dx}=2xy$ (B)…Preview
  82. Q82Family $y=Ax+A^3$ of curves will correspond to a differential equation of order: (A) 3 (B) 2 (C) 1 (D) not definedPreview
  83. Q83The general solution of $\frac{dy}{dx}=2x\,e^{x^2-y}$ is: (A) $e^{x^2-y}=c$ (B) $e^{-y}+e^{x^2}=c$ (C) $e^y=e^{x^2}+c$ (D) $e^{x^2}+y=c$Preview
  84. Q84The curve for which the slope of the tangent at any point is equal to the ratio of the abscissa to the ordinate of the point is: (A) an elli…Preview
  85. Q85The general solution of the differential equation $\frac{dy}{dx}=e^{\frac{x^2}{2}}+xy$ is: (A) $y=ce^{-\frac{x^2}{2}}$ (B) $y=ce^{\frac{x^2}…Preview
  86. Q86The solution of the equation $(2y-1)\,dx-(2x+3)\,dy=0$ is: (A) $\frac{2x-1}{2y+3}=k$ (B) $\frac{2y+1}{2x-3}=k$ (C) $\frac{2x+3}{2y-1}=k$ (D)…Preview
  87. Q87The differential equation for which $y=a\cos x+b\sin x$ is a solution, is: (A) $\frac{d^2y}{dx^2}+y=0$ (B) $\frac{d^2y}{dx^2}-y=0$ (C) $\fra…Preview
  88. Q88The solution of $\frac{dy}{dx}+y=e^{-x}$, $y(0)=0$ is: (A) $y=e^{-x}(x-1)$ (B) $y=xe^x$ (C) $y=xe^{-x}+1$ (D) $y=xe^{-x}$Preview
  89. Q89The order and degree of the differential equation $\left(\frac{d^2y}{dx^2}\right)^3-3\frac{d^3y}{dx^3}+2\left(\frac{dy}{dx}\right)^4=y^4$ ar…Preview
  90. Q90The order and degree of the differential equation $\left[1+\left(\frac{dy}{dx}\right)^2\right]=\frac{d^2y}{dx^2}$ are: (A) $2,\ \frac{3}{2}$…Preview
  91. Q91The differential equation of the family of curves $y^2=4a(x+a)$ is: (A) $y^2=4\frac{dy}{dx}\left(x+\frac{dy}{dx}\right)$ (B) $2y\frac{dy}{dx…Preview
  92. Q92Which of the following is the general solution of $\frac{d^2y}{dx^2}-2\frac{dy}{dx}+y=0$? (A) $y=(Ax+B)e^x$ (B) $y=(Ax+B)e^{-x}$ (C) $y=Ae^x…Preview
  93. Q93General solution of $\frac{dy}{dx}+y\tan x=\sec x$ is: (A) $y\sec x=\tan x+c$ (B) $y\tan x=\sec x+c$ (C) $\tan x=y\tan x+c$ (D) $x\sec x=\ta…Preview
  94. Q94Solution of the differential equation $\frac{dy}{dx}+\frac{y}{x}=\sin x$ is: (A) $x(y+\cos x)=\sin x+c$ (B) $x(y-\cos x)=\sin x+c$ (C) $xy\c…Preview
  95. Q95The general solution of the differential equation $(e^x+1)y\,dy=(y+1)e^x\,dx$ is: (A) $(y+1)=k(e^x+1)$ (B) $y+1=e^x+1+k$ (C) $y=\log\{k(y+1)…Preview
  96. Q96The solution of the differential equation $\frac{dy}{dx}=e^{x-y}+x^2 e^{-y}$ is: (A) $y=e^{x-y}-x^2 e^{-y}+c$ (B) $e^y-e^x=\frac{x^3}{3}+c$…Preview
  97. Q97The solution of the differential equation $\frac{dy}{dx}+\frac{2xy}{1+x^2}=\frac{1}{(1+x^2)^2}$ is: (A) $y(1+x^2)=c+\tan^{-1}x$ (B) $\frac{y…Preview

CBSE Sample Papers

Questions from official CBSE sample papers.

+Show 5 questions5 questions
  1. Q1The solution of the differential equation 𝑥𝑑𝑥 + 𝑦𝑑𝑦 = 0 represents a family of (A) straight lines (B) parabolas (C) Circles (D) EllipsesPreview
  2. Q2The value of ''n , such that the differential equation 𝒙𝒏 𝒅𝒚 𝒅𝒙 = 𝒚(𝒍𝒐𝒈𝒚 − 𝒍𝒐𝒈𝒙 + 𝟏); (𝐰𝐡𝐞𝐫𝐞 𝒙, 𝒚 ∈ 𝑹+) is homogeneous, is (A) 0 (B) 1 (C) 2…Preview
  3. Q3Which of the following is not a homogeneous function of $x$ and $y$ ? (A) $y^2 - xy$ (B) $x - 3y$ (C) $\sin^2 \frac{y}{x} + \frac{y}{x}$ (D)…Preview
  4. Q4Find the differential equation representing the family of curves $y = ae^{bx+5}$, where $a$ and $b$ are arbitrary constants.Preview
  5. Q5Find the particular solution of the differential equation $e^x \tan y\, dx + (2 - e^x)\sec^2 y\, dy = 0$, given that $y = \dfrac{\pi}{4}$ wh…Preview