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Mathematics · Class 12 Science

Ch 13Differential Equations — Class 12 Mathematics, concept-first.

In physics, chemistry and the other sciences, we constantly build mathematical models of situations where a rate of change is known — how fast a population grows, how fast a radioactive substance decays, how fast a body cools — and the real question is to recover the underlying quantity itself as a function of time (or…

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Order and Degree of a Differential Equation

A differential equation is any equation that contains at least one derivative — ordinary or partial — of an unknown function.

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Introduction

In physics, chemistry and the other sciences, we constantly build mathematical models of situations where a rate of change is known — how fast a population grows, how fast a radioactive substance deca…

6.1.2

Differential Equation

An equation which contains the derivative of a function is called a DIFFERENTIAL EQUATION. This definition is deliberately broad: it says nothing about which variable is independent, which is dependen…

6.2

Order and Degree of the Differential Equation

Two basic labels classify every differential equation: its ORDER and its DEGREE.

+Exercise 6.2i16 questions
  1. Q11Obtain the differential equation by eliminating the arbitrary constant: $y=4ax$Free
  2. Q12Obtain the differential equation by eliminating the arbitrary constant: $x^3+y^3=4ax$Free
  3. Q13Obtain the differential equation by eliminating the arbitrary constants: $Ax^2+By^2=1$Free
  4. Q14Obtain the differential equation by eliminating the arbitrary constants: $y=A\cos(\log x)+B\sin(\log x)$Preview
  5. Q15Obtain the differential equation by eliminating the arbitrary constant: $y=c^2+\dfrac{c}{x}$Preview
  6. Q16Obtain the differential equation by eliminating the arbitrary constants: $y=c_1e^{3x}+c_2e^{2x}$Preview
  7. Q17Obtain the differential equation by eliminating the arbitrary constant: $y=a+\dfrac{a}{x}$Preview
  8. Q18Obtain the differential equation by eliminating the arbitrary constants: $y=c_1e^{2x}+c_2e^{5x}$Preview
  9. Q19Obtain the differential equation by eliminating the arbitrary constants: $c_1x^3+c_2y^2=5$Preview
  10. Q20Obtain the differential equation by eliminating the arbitrary constants: $y=e^{-2x}(A\cos x+B\sin x)$Preview
  11. Q21Form the differential equation of the family of lines making intercept $a$ on the X-axis.Preview
  12. Q22Find the differential equation of all parabolas having length of latus rectum $4a$ and axis parallel to the X-axis.Preview
  13. Q23Find the differential equation of all ellipses whose major axis is twice its minor axis.Preview
  14. Q24Form the differential equation of the family of lines parallel to the line $2x+3y+4=0$.Preview
  15. Q25Find the differential equation of all circles having radius $9$ and centre at point $A(h,k)$.Preview
  16. Q26Form the differential equation of all parabolas whose axis is the X-axis.Preview
6.3

Formation of Differential Equation

A differential equation does not only arise by someone writing one down directly — very often it arises because a FAMILY of curves (or a physical relationship) is known, written with one or more ARBIT…

+Exercise 6.3i27 questions
  1. Q27Verify that $xy=\log y+c$ is a solution of $\dfrac{dy}{dx}=\dfrac{y^2}{1-xy}$.Free
  2. Q28Verify that $y=(\sin^{-1}x)^2+c$ is a solution of $(1-x^2)\dfrac{d^2y}{dx^2}-x\dfrac{dy}{dx}=2$.Free
  3. Q29Verify that $y=e^{-x}+Ax+B$ is a solution of $e^x\dfrac{d^2y}{dx^2}=1$.Free
  4. Q30Verify that $y=x^m$ is a solution of $x^2\dfrac{d^2y}{dx^2}-mx\dfrac{dy}{dx}+my=0$.Preview
  5. Q31Verify that $y=a+\dfrac{b}{x}$ is a solution of $x\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}=0$.Preview
  6. Q32Verify that $y=e^{ax}$ is a solution of $x\dfrac{dy}{dx}=y\log y$.Preview
  7. Q33Solve: $\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}$Preview
  8. Q34Solve: $\log\dfrac{dy}{dx}=2x+3y$Preview
  9. Q35Solve: $y-x\dfrac{dy}{dx}=0$Preview
  10. Q36Solve: $\sec^2x\cdot\tan y\,dx+\sec^2y\cdot\tan x\,dy=0$Preview
  11. Q37Solve: $\cos x\cdot\cos y\,dy-\sin x\cdot\sin y\,dx=0$Preview
  12. Q38Solve: $\dfrac{dy}{dx}=-k$, where $k=$ constant.Preview
  13. Q39Solve: $\dfrac{\cos^2y\cdot dy}{x}+\dfrac{\cos^2x\cdot dx}{y}=0$Preview
  14. Q40Solve: $y^3-\dfrac{dy}{dx}=x^2\dfrac{dy}{dx}$Preview
  15. Q41Solve: $2e^{x+2y}\cdot dx-3\,dy=0$Preview
  16. Q42Solve: $\dfrac{dy}{dx}=e^{x+y}+x^2e^y$Preview
  17. Q43Find the particular solution: $3e^x\tan y\cdot dx+(1+e^x)\sec^2y\cdot dy=0$, when $x=0,\ y=\pi$.Preview
  18. Q44Find the particular solution: $(x-y^2x)\cdot dx-(y+x^2y)\cdot dy=0$, when $x=2,\ y=0$.Preview
  19. Q45Find the particular solution: $y(1+\log x)\dfrac{dx}{dy}-x\log x=0,\ y=e^2,$ when $x=e$.Preview
  20. Q46Find the particular solution: $(e^y+1)\cos x+e^y\sin x\dfrac{dy}{dx}=0$, when $x=\dfrac{\pi}{6},\ y=0$.Preview
  21. Q47Find the particular solution: $(x+1)\dfrac{dy}{dx}-1=2e^{-y}$, $y=0,x=1$.Preview
  22. Q48Find the particular solution: $\cos\left(\dfrac{dy}{dx}\right)=a,\ a\in[1,2],\ y(0)=2$.Preview
  23. Q49Reduce to variable separable form and solve: $\dfrac{dy}{dx}=\cos(x+y)$Preview
  24. Q50Reduce to variable separable form and solve: $(x-y)^2\dfrac{dy}{dx}=a^2$Preview
  25. Q51Reduce to variable separable form and solve: $x+y\dfrac{dy}{dx}=\sec(x^2+y^2)$Preview
  26. Q52Reduce to variable separable form and solve: $\cos^2(x-2y)=1-2\dfrac{dy}{dx}$Preview
  27. Q53Reduce to variable separable form and solve: $(2x-2y+3)dx-(x-y+1)dy=0$, when $x=0,\ y=1$.Preview
6.4

Solution of Differential Equation

Once a differential equation is available, the next question is what it means to SOLVE it, and the simplest systematic technique for doing so — separation of variables.

+Exercise 6.4i15 questions
  1. Q54Solve: $x\sin\dfrac{y}{x}\,dy=\left[y\sin\dfrac{y}{x}-x\right]dx$Free
  2. Q55Solve: $(x^2-y^2)dx-2xy\cdot dy=0$Free
  3. Q56Solve: $\left(1+2e^{x/y}\right)+2e^{x/y}\left(1-\dfrac{x}{y}\right)\dfrac{dy}{dx}=0$Free
  4. Q57Solve: $y^2\cdot dx+(xy+x^2)dy=0$Preview
  5. Q58Solve: $(x^2-y^2)dx+2xy\cdot dy=0$Preview
  6. Q59Solve: $\dfrac{dy}{dx}+\dfrac{x-2y}{2x-y}=0$Preview
  7. Q60Solve: $x\dfrac{dy}{dx}-y+x\sin\dfrac{y}{x}=0$Preview
  8. Q61Solve: $\left(1+e^{x/y}\right)dx+e^{x/y}\left(1-\dfrac{x}{y}\right)dy=0$Preview
  9. Q62Solve: $y^2-x^2\dfrac{dy}{dx}=xy\dfrac{dy}{dx}$Preview
  10. Q63Solve: $xy\dfrac{dy}{dx}=x^2+2y^2$, $y(1)=0$Preview
  11. Q64Solve: $x\,dy+2y\cdot dx=0$, when $x=2,\ y=1$.Preview
  12. Q65Solve: $x^2\dfrac{dy}{dx}=x^2+xy+y^2$Preview
  13. Q66Solve: $(9x+5y)dy+(15x+11y)dx=0$Preview
  14. Q67Solve: $(x^2+3xy+y^2)dx-x^2\,dy=0$Preview
  15. Q68Solve: $(x^2+y^2)dx-2xy\cdot dy=0$Preview
6.4.1

Homogeneous Differential Equation

Recall that the DEGREE of a single term is the sum of the exponents (degrees) of all the variables it contains — for example, the term 3x²y²z has degree 2+2+1 = 5.

6.4.2

Linear Differential Equation

A differential equation of the type dy/dx + Py = Q, where P and Q are functions of x (or constants), is called a LINEAR DIFFERENTIAL EQUATION.

6.5.1

Population Growth and Growth of Bacteria

There are many real situations where the relation describing the RATE of change of a quantity is known directly from the physics or biology of the situation, and this rate description is itself a diff…

6.5.2

Radio Active Decay

Radioactive substances such as radium and caesium disintegrate over time — their mass decreases — and the RATE of disintegration is always proportional to the amount present at that time.

6.5.3

Newton's Law of Cooling

Newton's law of cooling states that the rate of change of the temperature of a heated (or cooled) body at any instant is proportional to the DIFFERENCE between the body's own temperature and the tempe…

6.5.4

Surface Area

Knowledge of a differential equation can also be used to solve problems involving surface area, where a rate of change of volume is described as being proportional to the wetted (or exposed) surface a…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 31 questions31 questions
  1. Q1A body is heated at $110°C$ and placed in air at $10°C$. After 1 hour its temperature is $60°C$. How much additional time is required for it…Preview
  2. Q2Solve the differential equation $\cos(x + y)\,dy = dx$. Hence find the particular solution for $x = 0$ and $y = 0$.Preview
  3. Q3Form the differential equation by eliminating arbitrary constants from the relation $y = Ae^{5x} + Be^{-5x}$Preview
  4. Q4Solve: $\dfrac{dy}{dx} = \cos(x + y)$Preview
  5. Q5If the population of a country doubles in 60 years, in how many years will it be triple under the assumption that the rate of increase is pr…Preview
  6. Q6Integrating factor of linear differential equation $x\dfrac{dy}{dx} + 2y = x^2 \log x$ is ______. (a) $\dfrac{1}{x^2}$ (b) $\dfrac{1}{x}$ (c…Preview
  7. Q7Obtain the differential equation by eliminating the arbitrary constants from the following equation: $y = c_1 e^{2x} + c_2 e^{-2x}$.Preview
  8. Q8Find the particular solution of the differential equation: $y(1+\log x)\dfrac{dx}{dy} - x\log x = 0$, when $y = e^2$ and $x = e$.Preview
  9. Q9If $y = a\mathrm{e}^{5x} + b\mathrm{e}^{-5x}$, then the differential equation is________. (a) $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=25y$ (b)…Preview
  10. Q10Solve the differential equation: $\dfrac{\mathrm{d}y}{\mathrm{d}x} + y\sec x = \tan x$ **OR** Solve the differential equation: $(x+y)\dfrac{…Preview
  11. Q11Order and degree of differential equation $\dfrac{d^4y}{dx^4} = \left[1+\left(\dfrac{dy}{dx}\right)^2\right]^3$ respectively are ________. (…Preview
  12. Q12Solve the differential equation $\dfrac{dy}{dx} = x^2y + y$Preview
  13. Q13Solve the differential equation $\dfrac{dy}{dx} + y = e^{-x}$Preview
  14. Q14Write the degree of the differential equation $(y''')^2 + 3(y'') + 3xy' + 5y = 0$Preview
  15. Q15Solve the differential equation $y\dfrac{dy}{dx} + x = 0$Preview
  16. Q16Form the differential equation of all lines which makes intercept 3 on x-axis.Preview
  17. Q17Solve $x + y\dfrac{dy}{dx} = \sec(x^2+y^2)$Preview
  18. Q18The solution of the differential equation $\dfrac{dx}{dt} = \dfrac{x\log x}{t}$ is ________. (a) $x = e^{ct}$ (b) $x + e^{ct} = 0$ (c) $x =…Preview
  19. Q19Write the degree of the differential equation $e^{\frac{dy}{dx}} + \dfrac{dy}{dx} = x$Preview
  20. Q20Solve the differential equation $\cos x \cos y\,dy - \sin x \sin y\,dx = 0$Preview
  21. Q21Find the particular solution of the differential equation $\dfrac{dy}{dx} = e^{2y}\cos x$, when $x = \dfrac{\pi}{6}$, $y = 0$.Preview
  22. Q22The integrating factor of linear differential equation $x\dfrac{dy}{dx}+2y=x^2\log x$ is ____. (a) $x$ (b) $\dfrac{1}{x}$ (c) $x^2$ (d) $\df…Preview
  23. Q23A particle is moving along X-axis. Its acceleration at time $t$ is proportional to its velocity at that time. Find the differential equation…Preview
  24. Q24Solve: $1+\dfrac{dy}{dx}=\csc(x+y)$; put $x+y=u$.Preview
  25. Q25Solve the differential equation: $x\cdot\dfrac{dy}{dx}-y+x\cdot\sin\left(\dfrac{y}{x}\right)=0$.Preview
  26. Q26Write the order of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\left(\dfrac{d^2y}{dx^2}\right)^{3/2}$Preview
  27. Q27Solve the differential equation: $x\dfrac{dy}{dx}=x\cdot\tan\left(\dfrac{y}{x}\right)+y$.Preview
  28. Q28If a body cools from $80°C$ to $50°C$ at room temperature of $25°C$ in 30 minutes, find the temperature of the body after 1 hour.Preview
  29. Q29Write the degree of the differential equation $(y''')^2+3(y'')^3+3xy'+5y=0$Preview
  30. Q30Solve the D.E. $3e^x\tan y\,dx+(1+e^x)\sec^2 y\,dy=0$.Preview
  31. Q31Solve the differential equation $x^2\cdot\dfrac{dy}{dx}=x^2+xy+y^2$.Preview

More questions

88 Q
+Show 10 questions10 questions
  1. Q1Determine the order and degree of the differential equation: $\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=2\sin x$Free
  2. Q2Determine the order and degree of the differential equation: $\sqrt[3]{1+\left(\dfrac{dy}{dx}\right)^2}=\dfrac{d^2y}{dx^2}$Free
  3. Q3Determine the order and degree of the differential equation: $\dfrac{dy}{dx}=\dfrac{2\sin x+3}{\dfrac{dy}{dx}}$Free
  4. Q4Determine the order and degree of the differential equation: $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}+x=\sqrt{1+\dfrac{d^3y}{dx^3}}$Preview
  5. Q5Determine the order and degree of the differential equation: $\dfrac{d^2y}{dt^2}+\left(\dfrac{dy}{dt}\right)^2+7x+5=0$Preview
  6. Q6Determine the order and degree of the differential equation: $(y''')^2+3y''+3xy'+5y=0$Preview
  7. Q7Determine the order and degree of the differential equation: $\left(\dfrac{d^2y}{dx^2}\right)^2+\cos\left(\dfrac{dy}{dx}\right)=0$Preview
  8. Q8Determine the order and degree of the differential equation: $\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}=8\dfrac{d^2y}{dx^2}$Preview
  9. Q9Determine the order and degree of the differential equation: $\left(\dfrac{d^3y}{dx^3}\right)^{1/2}-\left(\dfrac{dy}{dx}\right)^{1/3}=20$Preview
  10. Q10Determine the order and degree of the differential equation: $x+\dfrac{d^2y}{dx^2}=\sqrt{1+\left(\dfrac{d^2y}{dx^2}\right)^2}$Preview
+Show 15 questions15 questions
  1. Q69Solve: $\dfrac{dy}{dx}+\dfrac{y}{x}=x^3-3$Free
  2. Q70Solve: $\cos^2x\,\dfrac{dy}{dx}+y=\tan x$Free
  3. Q71Solve: $(x+2y^3)\dfrac{dy}{dx}=y$Free
  4. Q72Solve: $\dfrac{dy}{dx}+y\sec x=\tan x$Preview
  5. Q73Solve: $x\dfrac{dy}{dx}+2y=x^2\log x$Preview
  6. Q74Solve: $(x+y)\dfrac{dy}{dx}=1$Preview
  7. Q75Solve: $(x+a)\dfrac{dy}{dx}-3y=(x+a)^5$Preview
  8. Q76Solve: $dr+(2r\cot\theta+\sin2\theta)\,d\theta=0$Preview
  9. Q77Solve: $y\,dx+(x-y^2)dy=0$Preview
  10. Q78Solve: $(1-x^2)\dfrac{dy}{dx}+2xy=x(1-x^2)^{1/2}$Preview
  11. Q79Solve: $(1+x^2)\dfrac{dy}{dx}+y=e^{\tan^{-1}x}$Preview
  12. Q80Find the equation of the curve which passes through the origin and has slope $x+3y-1$ at any point $(x,y)$ on it.Preview
  13. Q81Find the equation of the curve passing through the point $\left(\dfrac{3}{\sqrt2},\sqrt2\right)$ having slope of the tangent at any point $(…Preview
  14. Q82A curve passes through the point $(0,2)$. The sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve…Preview
  15. Q83If the slope of the tangent to the curve at each of its points equals the sum of the abscissa and the product of the abscissa and ordinate,…Preview
+Show 12 questions12 questions
  1. Q84In a certain culture of bacteria the rate of increase is proportional to the number present. If it is found that the number doubles in 4 hou…Free
  2. Q85If the population of a country doubles in 60 years, in how many years will it be triple (treble) under the assumption that the rate of incre…Free
  3. Q86If a body cools from $80^\circ$c to $50^\circ$c at room temperature of $25^\circ$c in 30 minutes, find the temperature of the body after 1 h…Free
  4. Q87The rate of growth of bacteria is proportional to the number present. If initially there were 1000 bacteria and the number doubles in 1 hour…Preview
  5. Q88The rate of disintegration of a radioactive element at any time $t$ is proportional to its mass at that time. Find the time during which the…Preview
  6. Q89The rate of decay of a certain substance is directly proportional to the amount present at that instant. Initially there are 25 gms of the s…Preview
  7. Q90Find the population of a city at any time $t$, given that the rate of increase of population is proportional to the population at the instan…Preview
  8. Q91A body cools according to Newton's law from $100^\circ$c to $60^\circ$c in 20 minutes, with the surroundings at $20^\circ$c. How long will i…Preview
  9. Q92A right circular cone has height 9 cms and base radius 5 cms. It is inverted and water is poured into it. If at any instant the water level…Preview
  10. Q93Assume that a spherical raindrop evaporates at a rate proportional to its surface area. If its radius originally is 3mm and 1 hour later has…Preview
  11. Q94The rate of growth of the population of a city at any time $t$ is proportional to the size of the population. For a certain city the constan…Preview
  12. Q95Radium decomposes at a rate proportional to the amount present at any time. If $p$ percent of the amount disappears in one year, what percen…Preview
+Show 15 questions15 questions
  1. Q96The order and degree of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=\left(\dfrac{d^2y}{dx^2}\right)^{3/2}$ are respect…Free
  2. Q97The differential equation of $y=c^2+\dfrac{c}{x}$ is... (A) $x^4\left(\dfrac{dy}{dx}\right)^2-x\dfrac{dy}{dx}=y$ (B) $\dfrac{d^2y}{dx^2}+x\d…Free
  3. Q98$x^2+y^2=a^2$ is a solution of... (A) $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}-y=0$ (B) $y=x\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}+a^2y$ (C) $y=x\…Free
  4. Q99The differential equation of all circles having their centres on the line $y=5$ and touching the X-axis is... (A) $y^2\left(1+\dfrac{dy}{dx}…Preview
  5. Q100The differential equation $y\dfrac{dy}{dx}+x=0$ represents a family of... (A) circles (B) parabolas (C) ellipses (D) hyperbolasPreview
  6. Q101The solution of $\dfrac1x\cdot\dfrac{dy}{dx}=\tan^{-1}x$ is... (A) $\dfrac{x^2\tan^{-1}x}{2}+c=0$ (B) $x\tan^{-1}x+c=0$ (C) $x-\tan^{-1}x=c$…Preview
  7. Q102The solution of $(x+y)^2\dfrac{dy}{dx}=1$ is... (A) $x=\tan^{-1}(x+y)+c$ (B) $y\tan^{-1}\dfrac{x}{y}=c$ (C) $y=\tan^{-1}(x+y)+c$ (D) $y+\tan…Preview
  8. Q103The solution of $\dfrac{dy}{dx}=\dfrac{y+\sqrt{x^2-y^2}}{x}$ is... (A) $\sin^{-1}\dfrac{y}{x}=2\log|x|+c$ (B) $\sin^{-1}\dfrac{y}{x}=\log|x|…Preview
  9. Q104The solution of $\dfrac{dy}{dx}+y=\cos x-\sin x$ is... (A) $ye^x=\cos x+c$ (B) $ye^x+e^x\cos x=c$ (C) $ye^x=e^x\cos x+c$ (D) $y^2e^x=e^x\cos…Preview
  10. Q105The integrating factor of the linear differential equation $x\dfrac{dy}{dx}+2y=x^2\log x$ is... (A) $\dfrac1x$ (B) $k$ (C) $\dfrac{1}{x^2}$…Preview
  11. Q106The solution of the differential equation $\dfrac{dy}{dx}=\sec x-y\tan x$ is... (A) $y\sec x+\tan x=c$ (B) $y\sec x=\tan x+c$ (C) $\sec x+y\…Preview
  12. Q107The particular solution of $\dfrac{dy}{dx}=xe^{y-x}$, when $x=y=0$, is... (A) $e^{x-y}=x+1$ (B) $e^{x+y}=x+1$ (C) $e^x+e^y=x+1$ (D) $e^{y-x}…Preview
  13. Q108$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ is a solution of... (A) $\dfrac{d^2y}{dx^2}+yx+\left(\dfrac{dy}{dx}\right)^2=0$ (B) $xy\dfrac{d^2y}{dx…Preview
  14. Q109The decay rate of a certain substance is directly proportional to the amount present at that instant. Initially there are 27 grams and 3 hou…Preview
  15. Q110If the surrounding air is kept at $20^\circ$c and a body cools from $80^\circ$c to $70^\circ$c in 5 minutes, the temperature of the body aft…Preview
+Show 36 questions36 questions
  1. Q111Determine the order and degree: $\dfrac{d^2y}{dx^2}+5\dfrac{dy}{dx}+y=x^3$Free
  2. Q112Determine the order and degree: $\left(\dfrac{d^3y}{dx^3}\right)^2=\sqrt[5]{1+\dfrac{dy}{dx}}$Free
  3. Q113Determine the order and degree: $\sqrt[3]{1+\left(\dfrac{dy}{dx}\right)^2}=\dfrac{d^2y}{dx^2}$Free
  4. Q114Determine the order and degree: $\dfrac{dy}{dx}=3y+\sqrt[4]{1+5\left(\dfrac{dy}{dx}\right)^2}$Preview
  5. Q115Determine the order and degree: $\dfrac{d^4y}{dx^4}+\sin\!\left(\dfrac{dy}{dx}\right)=0$Preview
  6. Q116Verify: $x^2+y^2=r^2$, and $x\dfrac{dy}{dx}+r\sqrt{1+\left(\dfrac{dy}{dx}\right)^2}=y$Preview
  7. Q117Verify: $y=e^{ax}\sin bx$, and $\dfrac{d^2y}{dx^2}-2a\dfrac{dy}{dx}+(a^2+b^2)y=0$Preview
  8. Q118Verify: $y=3\cos(\log x)+4\sin(\log x)$, and $x^2\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}+y=0$Preview
  9. Q119Verify: $y=ae^x+be^{-x}+x^2$, and $x\dfrac{d^2y}{dx^2}+2\dfrac{dy}{dx}+x^3=xy+2$Preview
  10. Q120Verify: $x^2=2y^2\log y$, and $x^2+y^2=xy\dfrac{dx}{dy}$Preview
  11. Q121Obtain the differential equation by eliminating the arbitrary constants: $y^2=a(b-x)(b+x)$Preview
  12. Q122Obtain the differential equation by eliminating the arbitrary constants: $y=a\sin(x+b)$Preview
  13. Q123Obtain the differential equation by eliminating the arbitrary constants: $(y-a)^2=b(x+4)$Preview
  14. Q124Obtain the differential equation by eliminating the arbitrary constants: $y=\sqrt{a\cos(\log x)+b\sin(\log x)}$Preview
  15. Q125Obtain the differential equation by eliminating the arbitrary constants: $y=Ae^{3x+1}+Be^{-3x+1}$Preview
  16. Q126Form the differential equation of all circles which pass through the origin and whose centres lie on the X-axis.Preview
  17. Q127Form the differential equation of all parabolas which have $4b$ as latus rectum and whose axis is parallel to the Y-axis.Preview
  18. Q128Form the differential equation of all ellipses whose major axis is twice its minor axis.Preview
  19. Q129Form the differential equation of all the lines which are parallel to the line $3x-2y+7=0$.Preview
  20. Q130Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola $\dfra…Preview
  21. Q131Solve: $\log\dfrac{dy}{dx}=2x+3y$Preview
  22. Q132Solve: $\dfrac{dy}{dx}=x^2y+y$Preview
  23. Q133Solve: $\dfrac{dy}{dx}=\dfrac{2y-x}{2y+x}$Preview
  24. Q134Solve: $x\,dy=(x+y+1)\,dx$Preview
  25. Q135Solve: $\dfrac{dy}{dx}+y\cot x=x^2\cot x+2x$Preview
  26. Q136Solve: $y\log y=(\log y^2-x)\dfrac{dy}{dx}$Preview
  27. Q137Solve: $4\dfrac{dx}{dy}+8x=5e^{-3y}$Preview
  28. Q138Find the particular solution: $y(1+\log x)=(\log x^x)\dfrac{dy}{dx}$, when $y(e)=e^2$.Preview
  29. Q139Find the particular solution: $(x+2y^2)\dfrac{dy}{dx}=y$, when $x=2,\ y=1$.Preview
  30. Q140Find the particular solution: $\dfrac{dy}{dx}-3y\cot x=\sin2x$, when $y\left(\dfrac{\pi}{2}\right)=2$.Preview
  31. Q141Find the particular solution: $(x+y)dy+(x-y)dx=0$, when $x=1=y$.Preview
  32. Q142Find the particular solution: $2ye^{x/y}\,dx+(y-2xe^{x/y})\,dy=0$, when $y(0)=1$.Preview
  33. Q143Show that the general solution of the differential equation $\dfrac{dy}{dx}=\dfrac{y^2+y+1}{x^2+x+1}$ is given by $(x+y+1)=c(1-x-y-2xy)$.Preview
  34. Q144The normal lines to a given curve at each point $(x,y)$ on the curve pass through $(2,0)$. The curve passes through $(2,3)$. Find the equati…Preview
  35. Q145The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6…Preview
  36. Q146A person's assets start reducing so that the rate of reduction of assets is proportional to the square root of the assets existing at that m…Preview