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Mathematics · Class 12 Science

Ch 10Ordinary Differential Equations — Class 12 Mathematics, concept-first.

Many real situations — the motion of a projectile, rocket, satellite or planet; the current in an electric circuit; the conduction of heat along a rod; the vibration of a wire or membrane — force us to describe how a quantity's rate of change (its derivative) is related to the quantity itself and to other variables.

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Key concepts

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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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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

10.1

Introduction

Many real situations — the motion of a projectile, rocket, satellite or planet; the current in an electric circuit; the conduction of heat along a rod; the vibration of a wire or membrane — force us t…

10.2

Differential Equation, Order, and Degree

Definition 10.1 (Differential equation). A differential equation is any equation containing at least one derivative of an unknown function — either an ordinary derivative (one independent variable) or…

+Exercise 10.1i1 question
  1. Q1For each of the following differential equations, determine its order, degree (if it exists): (i) $\dfrac{dy}{dx}+xy=\cot x$ (ii) $\left(\df…Preview
10.3

Classification of Differential Equations

Definition 10.4 (Ordinary Differential Equation). If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable, it is an Ordinar…

10.4

Formation of Differential Equations

Differential equations that model real problems do not, in practice, arise by starting from a family of curves and eliminating constants — they arise directly from a physical law or geometric conditio…

10.4.1

Formation of Differential Equations from Physical Situations

Physical laws that state how a quantity's rate of change is related to other quantities translate immediately into a differential equation — no arbitrary constants need to be eliminated, because the l…

+Exercise 10.2i2 questions
  1. Q1Express each of the following physical statements in the form of a differential equation. (i) Radium decays at a rate proportional to the am…Free
  2. Q2Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of…Preview
10.4.2

Formation of Differential Equations from Geometrical Problems

When a family of curves is given by an equation containing arbitrary constants, the differential equation the whole family satisfies is found by differentiating times and eliminating the constants fro…

+Exercise 10.3i8 questions
  1. Q1Find the differential equation of the family of (i) all non-vertical lines in a plane (ii) all non-horizontal lines in a plane.Free
  2. Q2Form the differential equation of all straight lines touching the circle $x^2+y^2=r^2$.Free
  3. Q3Find the differential equation of the family of circles passing through the origin and having their centres on the $x$-axis.Free
  4. Q4Find the differential equation of the family of all the parabolas with latus rectum $4a$ and whose axes are parallel to the $x$-axis.Preview
  5. Q5Find the differential equation of the family of parabolas with vertex at $(0,-1)$ and having axis along the $y$-axis.Preview
  6. Q6Find the differential equations of the family of all the ellipses having foci on the $y$-axis and centre at the origin.Preview
  7. Q7Find the differential equation corresponding to the family of curves represented by the equation $y=\mathrm{A}e^{8x}+\mathrm{B}e^{-8x}$, whe…Preview
  8. Q8Find the differential equation of the curve represented by $xy=ae^x+be^{-x}+x^2$.Preview
10.5

Solution of Ordinary Differential Equations

Definition 10.9 (Solution of a DE). A solution of a differential equation is an expression for the dependent variable in terms of the independent variable(s) that satisfies the equation when substitut…

+Exercise 10.4i8 questions
  1. Q1Show that each of the following expressions is a solution of the corresponding given differential equation. (i) $y=2x^2$ ; $xy'=2y$ (ii) $y=…Free
  2. Q2Find the value of $m$ so that the function $y=e^{mx}$ is a solution of the given differential equation. (i) $y'+2y=0$ (ii) $y''-5y'+6y=0$Free
  3. Q3The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through $(2,…Free
  4. Q4Show that $y=e^{-x}+mx+n$ is a solution of the differential equation $e^x\left(\dfrac{d^2y}{dx^2}\right)-1=0$.Preview
  5. Q5Show that $y=ax+\dfrac{b}{x},\ x\ne 0$ is a solution of the differential equation $x^2y''+xy'-y=0$.Preview
  6. Q6Show that $y=ae^{-3x}+b$, where $a$ and $b$ are arbitrary constants, is a solution of the differential equation $\dfrac{d^2y}{dx^2}+3\dfrac{…Preview
  7. Q7Show that the differential equation representing the family of curves $y^2=2a\left(x+a^{2/3}\right)$, where $a$ is a positive parameter, is…Preview
  8. Q8Show that $y=a\cos bx$ is a solution of the differential equation $\dfrac{d^2y}{dx^2}+b^2y=0$.Preview
10.6

Solution of First Order and First Degree Differential Equations

Having seen how a differential equation is formed and what it means to solve one, the chapter now develops concrete methods for actually solving first-order, first-degree differential equations — equa…

10.6.1

Variables Separable Method

Separation of variables — the oldest and most direct method for solving a first-order equation — was introduced by Leibniz and later formalised by John Bernoulli in 1694.

10.6.2

Substitution Method

When an equation has the shape — a function of the single linear combination , rather than of and separately — it is not immediately separable, but a linear substitution converts it into an equation t…

+Exercise 10.5i4 questions
  1. Q1If $F$ is the constant force generated by the motor of an automobile of mass $M$, its velocity $V$ is given by $M\dfrac{dV}{dt}=F-kV$, where…Free
  2. Q2The velocity $v$, of a parachute falling vertically satisfies the equation $v\dfrac{dv}{dx}=g\left(1-\dfrac{v^2}{k^2}\right)$, where $g$ and…Free
  3. Q3Find the equation of the curve whose slope is $\dfrac{y-1}{x^2+x}$ and which passes through the point $(1,0)$.Preview
  4. Q4Solve the following differential equations: (i) $\dfrac{dy}{dx}=\sqrt{\dfrac{1-y^2}{1-x^2}}$ (ii) $y\,dx+\left(1+x^2\right)\tan^{-1}x\,dy=0$…Preview
10.6.3

Homogeneous Form or Homogeneous Differential Equation

Definition 10.12 (Homogeneous function of degree ). A function is homogeneous of degree if for all suitably restricted (this scaling property is called Euler's homogeneity).

+Exercise 10.6i8 questions
  1. Q1$\left[x+y\cos\!\left(\dfrac{y}{x}\right)\right]dx=x\cos\!\left(\dfrac{y}{x}\right)dy$Free
  2. Q2$\left(x^3+y^3\right)dy-x^2y\,dx=0$Free
  3. Q3$ye^{x/y}dx=\left(xe^{x/y}+y\right)dy$Free
  4. Q4$2xy\,dx+\left(x^2+2y^2\right)dy=0$Preview
  5. Q5$\left(y^2-2xy\right)dx=\left(x^2-2xy\right)dy$Preview
  6. Q6$x\dfrac{dy}{dx}=y-x\cos^2\!\left(\dfrac{y}{x}\right)$Preview
  7. Q7$\left(1+3e^{y/x}\right)dy+3e^{y/x}\left(1-\dfrac{y}{x}\right)dx=0$, given that $y=0$ when $x=1$Preview
  8. Q8$\left(x^2+y^2\right)dy=xy\,dx$. It is given that $y(1)=1$ and $y(x_0)=e$. Find the value of $x_0$.Preview
10.7

First Order Linear Differential Equations

Definition. A first-order differential equation of the form — where and are functions of only — is called linear: no product of with occurs, and together with its derivative occur only to the first de…

+Exercise 10.7i15 questions
  1. Q1$\cos x\dfrac{dy}{dx}+y\sin x=1$Free
  2. Q2$\left(1-x^2\right)\dfrac{dy}{dx}-xy=1$Free
  3. Q3$\dfrac{dy}{dx}+\dfrac{y}{x}=\sin x$Free
  4. Q4$\left(x^2+1\right)\dfrac{dy}{dx}+2xy=\sqrt{x^2+4}$Preview
  5. Q5$\left(2x-10y^3\right)dy+y\,dx=0$Preview
  6. Q6$x\sin x\dfrac{dy}{dx}+\left(x\cos x+\sin x\right)y=\sin x$Preview
  7. Q7$\left(y-e^{\sin^{-1}x}\right)\dfrac{dx}{dy}+\sqrt{1-x^2}=0$Preview
  8. Q8$\dfrac{dy}{dx}+\dfrac{y}{(1-x)\sqrt{x}}=1-\sqrt{x}$Preview
  9. Q9$\left(1+x+xy^2\right)\dfrac{dy}{dx}+\left(y+y^3\right)=0$Preview
  10. Q10$\dfrac{dy}{dx}+\dfrac{y}{x\log x}=\dfrac{\sin 2x}{\log x}$Preview
  11. Q11$\left(x+a\right)\dfrac{dy}{dx}-2y=\left(x+a\right)^4$Preview
  12. Q12$\dfrac{dy}{dx}=\dfrac{\sin^2x}{1+x^3}-\dfrac{3x^2}{1+x^3}y$Preview
  13. Q13$x\dfrac{dy}{dx}+y=x\log x$Preview
  14. Q14$x\dfrac{dy}{dx}+2y-x^2\log x=0$Preview
  15. Q15$\dfrac{dy}{dx}+\dfrac{3y}{x}=\dfrac{1}{x^2}$, given that $y=2$ when $x=1$Preview
10.8

Applications of First Order Ordinary Differential Equations

Differential equations model real-world change: their solutions predict how a system will behave at a future time or in an unknown location.

10.8.1

Population growth

Consider the growth of a population — human, animal, or a bacteria colony — as a function of time . Let denote the population size at time ; although genuinely integer-valued, is approximated as a dif…

10.8.2

Radioactive decay

An atomic nucleus consists of protons and neutrons; many such combinations are unstable and spontaneously decay (transmute) into another substance — such nuclei are radioactive.

10.8.3

Newton's Law of cooling/warming

Pour a cup of coffee onto a table in an room, and it cools until it reaches room temperature; take a glass of water from the fridge into the same room, and it warms up until it reaches room temperatur…

10.8.4

Mixture problems

Mixing problems occur constantly in the chemical industry, and a single bookkeeping idea handles them all.

+Exercise 10.8i10 questions
  1. Q1The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the number tr…Free
  2. Q2Find the population of a city at any time $t$, given that the rate of increase of population is proportional to the population at that insta…Free
  3. Q3The equation of electromotive force for an electric circuit containing resistance and self-inductance is $E=Ri+L\dfrac{di}{dt}$, where $E$ i…Free
  4. Q4The engine of a motor boat moving at $10\ m/s$ is shut off. Given that the retardation at any subsequent time (after shutting off the engine…Preview
  5. Q5Suppose a person deposits $\textrm{\textrupee}\,10{,}000$ in a bank account at the rate of $5\%$ per annum compounded continuously. How much…Preview
  6. Q6Assume that the rate at which radioactive nuclei decay is proportional to the number of such nuclei that are present in a given sample. In a…Preview
  7. Q7Water at temperature $100^\circ C$ cools in $10$ minutes to $80^\circ C$ in a room temperature of $25^\circ C$. Find (i) The temperature of…Preview
  8. Q8At 10.00 A.M. a woman took a cup of hot instant coffee from her microwave oven and placed it on a nearby kitchen counter to cool. At this in…Preview
  9. Q9A pot of boiling water at $100^\circ C$ is removed from a stove at time $t=0$ and left to cool in the kitchen. After $5$ minutes, the water…Preview
  10. Q10A tank initially contains $50$ litres of pure water. Starting at time $t=0$ a brine containing $2$ grams of dissolved salt per litre flows i…Preview
10.9

Choose the Correct Answer

This exercise revises the whole chapter through multiple-choice questions: classifying order and degree (including after clearing a fractional power, or recognising when the degree is not defined); fo…

+Exercise 10.9i25 questions
  1. Q1The order and degree of the differential equation $\dfrac{d^2y}{dx^2}+\left(\dfrac{dy}{dx}\right)^{1/3}+x^{1/4}=0$ are respectively (1) $2,\…Free
  2. Q2The differential equation representing the family of curves $y=\mathrm{A}\cos(x+\mathrm{B})$, where $\mathrm{A}$ and $\mathrm{B}$ are parame…Free
  3. Q3The order and degree of the differential equation $\sqrt{\sin x}\left(dx+dy\right)=\sqrt{\cos x}\left(dx-dy\right)$ is (1) $1,\ 2$ (2) $2,\…Free
  4. Q4The order of the differential equation of all circles with centre at $(h,k)$ and radius '$a$' is (1) $2$ (2) $3$ (3) $4$ (4) $1$Preview
  5. Q5The differential equation of the family of curves $y=\mathrm{A}e^x+\mathrm{B}e^{-x}$, where $\mathrm{A}$ and $\mathrm{B}$ are arbitrary cons…Preview
  6. Q6The general solution of the differential equation $\dfrac{dy}{dx}=\dfrac{y}{x}$ is (1) $xy=k$ (2) $y=k\log x$ (3) $y=kx$ (4) $\log y=kx$Preview
  7. Q7The solution of the differential equation $2x\dfrac{dy}{dx}-y=3$ represents (1) straight lines (2) circles (3) parabola (4) ellipsePreview
  8. Q8The solution of $\dfrac{dy}{dx}+p(x)y=0$ is (1) $y=ce^{\int p\,dx}$ (2) $y=ce^{-\int p\,dx}$ (3) $x=ce^{-\int p\,dy}$ (4) $x=ce^{\int p\,dy}…Preview
  9. Q9The integrating factor of the differential equation $\dfrac{dy}{dx}+y=\dfrac{1+y}{\lambda}$ is (1) $\dfrac{x}{e^{\lambda}}$ (2) $\dfrac{e^{\…Preview
  10. Q10The integrating factor of the differential equation $\dfrac{dy}{dx}+P(x)y=Q(x)$ is $x$, then $P(x)$ (1) $x$ (2) $\dfrac{x^2}{2}$ (3) $\dfrac…Preview
  11. Q11The degree of the differential equation $y(x)=1+\dfrac{dy}{dx}+\dfrac{1}{1\cdot 2}\left(\dfrac{dy}{dx}\right)^2+\dfrac{1}{1\cdot 2\cdot 3}\l…Preview
  12. Q12If $p$ and $q$ are the order and degree of the differential equation $y\dfrac{dy}{dx}+x^3\left(\dfrac{d^2y}{dx^2}\right)+xy=\cos x$, when (1…Preview
  13. Q13The solution of the differential equation $\dfrac{dy}{dx}+\dfrac{1}{\sqrt{1-x^2}}=0$ is (1) $y+\sin^{-1}x=c$ (2) $x+\sin^{-1}y=0$ (3) $y^2+2…Preview
  14. Q14The solution of the differential equation $\dfrac{dy}{dx}=2xy$ is (1) $y=Ce^{x^2}$ (2) $y=2x^2+C$ (3) $y=Ce^{-x^2}+C$ (4) $y=x^2+C$Preview
  15. Q15The general solution of the differential equation $\log\!\left(\dfrac{dy}{dx}\right)=x+y$ is (1) $e^x+e^y=C$ (2) $e^x+e^{-y}=C$ (3) $e^{-x}+…Preview
  16. Q16The solution of $\dfrac{dy}{dx}=2^{y-x}$ is (1) $2^x+2^y=C$ (2) $2^x-2^y=C$ (3) $\dfrac{1}{2^x}-\dfrac{1}{2^y}=C$ (4) $x+y=C$Preview
  17. Q17The solution of the differential equation $\dfrac{dy}{dx}=\dfrac{y}{x}+\dfrac{\phi\!\left(\frac{y}{x}\right)}{\phi'\!\left(\frac{y}{x}\right…Preview
  18. Q18If $\sin x$ is the integrating factor of the linear differential equation $\dfrac{dy}{dx}+Py=Q$, then $P$ is (1) $\log\sin x$ (2) $\cos x$ (…Preview
  19. Q19The number of arbitrary constants in the general solutions of order $n$ and $n+1$ are respectively (1) $n-1,\ n$ (2) $n,\ n+1$ (3) $n+1,\ n+…Preview
  20. Q20The number of arbitrary constants in the particular solution of a differential equation of third order is (1) $3$ (2) $2$ (3) $1$ (4) $0$Preview
  21. Q21Integrating factor of the differential equation $\dfrac{dy}{dx}=\dfrac{x+y+1}{x+1}$ is (1) $\dfrac{1}{x+1}$ (2) $x+1$ (3) $\dfrac{1}{\sqrt{x…Preview
  22. Q22The population $P$ in any year $t$ is such that the rate of increase in the population is proportional to the population. Then (1) $P=Ce^{kt…Preview
  23. Q23$P$ is the amount of a certain substance left after time $t$. If the rate of evaporation of the substance is proportional to the amount rema…Preview
  24. Q24If the solution of the differential equation $\dfrac{dy}{dx}=\dfrac{ax+3}{2y+f}$ represents a circle, then the value of $a$ is (1) $2$ (2) $…Preview
  25. Q25The slope at any point of a curve $y=f(x)$ is given by $\dfrac{dy}{dx}=3x^2$ and it passes through $(-1,1)$. Then the equation of the curve…Preview
10.10

Summary

Chapter recap.

Sample & Board Papers

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

+Show 45 questions45 questions
  1. Q1The particular integral of the differential equation $f(D)y=e^{ax}$ where $f(D)=(D-a)g(D)$, $g(a)\neq 0$ is : (a) $m\,e^{ax}$ (b) $\dfrac{e^…Preview
  2. Q2Identify the incorrect statement : (a) The order of a differential equation is the order of the highest order derivative occurring in it. (b…Preview
  3. Q3The degree of the differential equation $\sqrt{1+\left(\dfrac{dy}{dx}\right)^{1/3}} = \dfrac{d^2y}{dx^2}$ is : (a) $1$ (b) $2$ (c) $3$ (d) $…Preview
  4. Q4The differential equation satisfied by all the straight lines in $xy$-plane (not parallel to $y$-axis) is : (a) $\dfrac{dy}{dx} = $ a consta…Preview
  5. Q5The temperature T of a cooling object drops at a rate proportional to the difference $(T-S)$, where S is constant temperature of surrounding…Preview
  6. Q6Solve : $(x^2+y^2)\,dx + 3xy\,dy = 0$Preview
  7. Q7Solve the differential equation $(36D^2-24D+13)y = 2\sin^2 x - e^{-x} + 2$.Preview
  8. Q8The differential equation $\left(\dfrac{dx}{dy}\right)^2 + 5y^{\frac{1}{3}} = x$ is : (a) of order 2 and degree 1 (b) of order 1 and degree…Preview
  9. Q9Solution of $\dfrac{dx}{dy} + mx = 0$, where $m < 0$ is : (a) $x = ce^{my}$ (b) $x = ce^{-my}$ (c) $x = my + c$ (d) $x = c$Preview
  10. Q10If $y = ke^{\lambda x}$ then its differential equation is (where k is arbitrary constant) : (a) $\dfrac{dy}{dx} = \lambda y$ (b) $\dfrac{dy}…Preview
  11. Q11The integrating factor of the differential equation $\dfrac{dy}{dx} - y\tan x = \cos x$ is : (a) $\sec x$ (b) $\cos x$ (c) $e^{\tan x}$ (d)…Preview
  12. Q12Solve : $(D^2 - 4D + 1)y = x^2$Preview
  13. Q13A cup of coffee at temperature $100°C$ is placed in a room whose temperature is $15°C$ and it cools to $60°C$ in 5 minutes. Find its tempera…Preview
  14. Q14The order and degree of the differential equation $y' + (y'')^2 = (x + y'')^2$ are : (a) 2, 1 (b) 1, 1 (c) 2, 2 (d) 1, 2Preview
  15. Q15The value of 'a' so that the curves $y = 3e^x$ and $y = \dfrac{a}{3}e^{-x}$ intersect orthogonally is : (a) $\dfrac{1}{3}$ (b) $-1$ (c) $3$…Preview
  16. Q16If $\cos x$ is an integrating factor of the differential equation $\dfrac{dy}{dx} + Py = Q$ then $P =$ (a) $\tan x$ (b) $-\cot x$ (c) $-\tan…Preview
  17. Q17The differential equation of all circles with centre at the origin is : (a) $x\,dx + y\,dy = 0$ (b) $x\,dy + y\,dx = 0$ (c) $x\,dx - y\,dy =…Preview
  18. Q18If $\dfrac{dy}{dx} = \dfrac{x - y}{x + y}$ then : (a) $x^2 + y^2 - 2xy = c$ (b) $2xy + y^2 + x^2 = c$ (c) $x^2 - y^2 - 2xy = c$ (d) $x^2 + y…Preview
  19. Q19Solve: $(2D^2 + 5D + 2)y = e^{-\frac{1}{2}x}$Preview
  20. Q20Solve: $(x^3 + 3xy^2)dx + (y^3 + 3x^2y)dy = 0$Preview
  21. Q21The rate at which the population of a city increases at any time is proportional to the population at that time. If there were 1,30,000 peop…Preview
  22. Q22$y = cx - c^2$ is the general solution of the differential equation : (a) $y' = c$ (b) $(y')^2 + xy' + y = 0$ (c) $(y')^2 - xy' + y = 0$ (d)…Preview
  23. Q23The order and degree of the differential equation $y' + (y'')^2 = x(x + y'')^2$ are : (a) $1, 2$ (b) $1, 1$ (c) $2, 2$ (d) $2, 1$Preview
  24. Q24Show that the solution of the differential equation $yx^3 dx + e^{-x} dy = 0$ is $(x^3 - 3x^2 + 6x - 6)e^x + \log y = c$.Preview
  25. Q25The order of the differential equation of all circles with centre at (h, k) and radius 'a', where h, k and a are arbitrary constants, is : (…Preview
  26. Q26The order and degree of the differential equation $\dfrac{dx}{dy}+\dfrac{dy}{dx}=0$ are : (a) 2, degree not defined (b) 1, 2 (c) 2, 1 (d) 2,…Preview
  27. Q27Find the differential equation of the family of parabolas $y^2=4ax$, where 'a' is an arbitrary constant.Preview
  28. Q28(a) In an investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective also measured the body temperature a…Preview
  29. Q29The solution of $\dfrac{dy}{dx}+p(x)y=0$ is : (a) $x=ce^{-\int p\,dy}$ (b) $y=ce^{\int p\,dx}$ (c) $x=ce^{\int p\,dy}$ (d) $y=ce^{-\int p\,d…Preview
  30. Q30If $\sin x$ is the integrating factor of the linear differential equation $\dfrac{dy}{dx}+Py=Q$, then P is : (a) $\tan x$ (b) $\log \sin x$…Preview
  31. Q31Show that the differential equation of the family of curves $y=Ae^x+Be^{-x}$, where A and B are arbitrary constants, is $\dfrac{d^2y}{dx^2}-…Preview
  32. Q32Solve : $\dfrac{dy}{dx}=\dfrac{\sqrt{1-y^2}}{\sqrt{1-x^2}}$Preview
  33. Q33(a) Show that the solution of the differential equation $(1+x^2)\dfrac{dy}{dx}=1+y^2$ is $\tan^{-1}y=\tan^{-1}x+C$ (or) $\tan^{-1}x=\tan^{-1…Preview
  34. Q34The general solution of the differential equation $\dfrac{dy}{dx}=\dfrac{y}{x}$ is : (a) $y=kx$ (b) $xy=k$ (c) $\log y=kx$ (d) $y=k\log x$Preview
  35. Q35Solve : $x\cos y\,dy=e^x(x\log x+1)\,dx$Preview
  36. Q36The differential equation of the family of curves $y=Ae^x+Be^{-x}$, where A and B are arbitrary constants is : (a) $\dfrac{dy}{dx}+y=0$ (b)…Preview
  37. Q37Find the differential equation for the family of all straight lines passing through the origin.Preview
  38. Q38Solve : $(1+x^2)\dfrac{dy}{dx}=1+y^2$Preview
  39. Q39(a) The rate of increase in the number of bacteria in a certain bacteria culture is proportional to the number present. Given that the numbe…Preview
  40. Q40The order and degree of the differential equation $\sqrt{\dfrac{dy}{dx}}-4\dfrac{dy}{dx}-7x=0$ are respectively : (a) $1, 2$ (b) $2, 1$ (c)…Preview
  41. Q41The slope at any point of a curve $y=f(x)$ is given by $\dfrac{dy}{dx}=3x^2$ and it passes through $(-1, 1)$. Then the equation of the curve…Preview
  42. Q42Solve $\dfrac{dy}{dx}=\sqrt{\dfrac{1-y^2}{1-x^2}}$Preview
  43. Q43The solution of the differential equation $2x\dfrac{dy}{dx}-y=3$ represents : (a) Parabola (b) Straight lines (c) Ellipse (d) CirclesPreview
  44. Q44P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaini…Preview
  45. Q45Determine the order and degree (if exists) of the differential equation $x^2\dfrac{d^2y}{dx^2}+\left[1+\left(\dfrac{dy}{dx}\right)^2\right]^…Preview