Differential Equations

Previous Years' CBSE Board Questions

9.2 Basic Concepts

MCQ

The sum of the order and the degree of the differential equation <math>\frac{d}{dx} \left( \left( \frac{dy}{dx} \right)^3 \right)</math> is

1. 2<br>(c) 5 (c) (b)<br>(d) 3

(2023)

1. The order and the degree of the differential equation <math>\left(1+3\frac{dy}{dx}\right)^2 = 4\frac{d^3y}{dx^3}</math> respectively are
2. 1, = (b) 3, 1 (c) 3, 3 (d) 1, 2

(2023)

VSA (1 mark)

1. The degree of the differential equation <math>1 + \left(\frac{dy}{dx}\right)^2 = x</math> is _____ (2020)
2. Find the order and the degree of the differential equation <math>x^2 \frac{d^2y}{dx^2} = \left\{1 + \left(\frac{dy}{dx}\right)^2\right\}^4</math>. (Delhi 2019) 🕕
3. Write the sum of the order and degree of the following differential equation <math>\frac{d}{dx}\left\{\left(\frac{dy}{dx}\right)^3\right\}=0.</math> (AI 2015)
4. Write the sum of the order and degree of the differential equation <math>\left(\frac{d^2y}{dx^2}\right)^2 + \left(\frac{dy}{dx}\right)^3 + x^4 = 0</math>. (Foreign 2015) U

Write the sum of the order and degree of the differential equation <math>1+\left(\frac{dy}{dx}\right)^4=7\left(\frac{d^2y}{dx^2}\right)^3</math>. (Delhi 2015C)

SAI (2 marks)

1. Find the product of the order and the degree of the differential equation <math>\left| \frac{d}{dx} (xy^2) \right| \cdot \frac{dy}{dx} + y = 0</math>.

(2022) U

(Delhi 2015, Al 2015C) [II

Find the value of (2a - 3b), if a and b represent respectively the order and the degree of the differential equation <math>x \left[ y \left( \frac{d^2 y}{dx^2} \right)^3 + x \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} \frac{dy}{dx} \right] = 0.</math>

(2022 C)

9.3 General and Particular Solutions of a Differential Equation

MCQ

The number of solutions of the differential equation

<math>\frac{dy}{dy} = \frac{y+1}{y+1}</math> when <math>y(1) = 2</math>, is

1. zero (b) one (c) two (d) infinite (2023)

The number of arbitrary constants in the particular

solution of a differential equation of second order is

(are)

1. 0 (b) 1 (c) 2 (d) 3 (2020) R

9.4 Methods of Solving First Order, First Degree Differential Equations

MCQ

The integrating factor for solving the differential

equation <math>x \frac{dy}{dx} - y = 2x^2</math> is

(a) <math>e^{-y}</math> (b) <math>e^{-e}</math> (c) x (d) <math>\frac{1}{y}</math>

(2023)

13. The integrating factor of the differential equation <math>(x+3y^2)\frac{dy}{dx} = y</math> is<br>

(a) y (b) -y (c) <math>\frac{1}{y}</math> (d) <math>-\frac{1}{y}</math>

(2020)

VSA (1 mark)

1. The integrating factor of the differential equation <math>x \frac{dy}{dx} - y = \log x</math> is _____. (2020 C) U
2. The integrating factor of the differential equation <math>x \frac{dy}{dx} + 2y = x^2</math> is ______ (2020)

Find the general solution of the differential equation

<math>e^{y-x}\frac{dy}{dx}=1.</math> (2020)

<math display="block">\textbf{17}. \ \ \mathsf{Find} \ \mathsf{the} \ \mathsf{integrating} \ \mathsf{factor} \ \mathsf{of} \ \mathsf{the} \ \mathsf{differential} \ \mathsf{equation}</math>

<math display="block">\left(\frac{e^{-2\sqrt{x}}}{\sqrt{x}} - \frac{y}{\sqrt{x}}\right) \frac{dx}{dy} = 1.</math>

18. Write the integrating factor of the following

differential equation:

<math>(1+y^2)+(2xy-\cot y)\frac{dy}{dx}=0</math> (Al 2015)

Write the solution of the differential equation <math>\frac{dy}{dx} = 2^{-y}</math>. (Foreign 2015) Ap

1. Find the solution of the differential equation <math>\frac{dy}{dy} = x^3 e^{-2y}</math>. (AI 2015C)

SAI (2 marks)

1. Find the general solution of the differential equation: <math>\log\left(\frac{dy}{dx}\right) = ax + by</math>. (Term II, 2021-22) (Ap)

11 Find the general solution of the differential equation <math>\sec^2 x \cdot \tan y \, dx + \sec^2 y \cdot \tan x \, dy = 0.</math> (Term II, 2021-22)

1. Find the general solution of the following differential equation: <math>\frac{dy}{dx} = e^{x-y} + x^2 e^{-y}</math> (Term II, 2021-22) (Ap)
2. Find the integrating factor of <math>x \frac{dy}{dx} + (1 + x \cot x)y = x</math>. (2021 C)
3. Solve the following homogeneous differential equation: <math>x \frac{dy}{dx} = x + y</math> (2020 C)

(Term II, 2021-22C)

1. Solve the following differential equation: <math>\frac{dy}{dx} + y = \cos x - \sin x</math> (Al 2019)

SAII (3 marks)

Find the particular solution of the differential equation <math>\frac{dy}{dx} = \frac{x+y}{x}</math>, <math>y(1)=0</math>. (2023)

1. Find the general solution of the differential equation

<math>e^x \tan y dx + (1 - e^x) \sec^2 y dy = 0.</math> (2023)

Find the particular solution of the differential equation <math>x \frac{dy}{dx} + x \cos^2(\frac{y}{x}) = y</math>; given that when <math>x=1, y=\frac{\pi}{4}</math>. (Term II, 2021-22) (hij

1. Find the general solution of the differential equation <math>x \frac{dy}{dx} = y (\log y - \log x + 1).</math> (Term II, 2021-22)
2. If the solution of the differential equation <math display="block">\frac{dy}{dx} = \frac{2xy - y^2}{2x^2} \text{ is } \frac{ax}{y} = b \log|x| + C, \text{ find the value of } a</math> and b. (2021C) Ap

(4 marks)

Case study: An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation A differential equation of the form <math>\frac{dy}{dx} = F(x,y)</math> is said

to be homogeneous if <math>F(x, y)</math> is a homogeneous

function of degree zero, whereas a function <math>F(x, y)</math>

is a homogenous function of degree n if <math>F(\lambda x , \lambda y)</math>

λⁿF(x, y). To solve a homogeneous differential

equation of the type <math>\frac{dy}{dy} = F(x,y) = g\left(\frac{y}{y}\right)</math> we make

substitution <math>y = vx</math> and then separate the variables.

Based on the above, answer the following questions.

Show that <math>(x^2 - y^2) dx + 2xy dy = 0</math> is a differential

equation of the type <math>\frac{dy}{dx} = g\left(\frac{y}{x}\right)</math>

Solve the above equation to find its general

solution. (2023)

Find the particular solution of the differential

equation <math>(1+x^2)\frac{dy}{dx} + 2xy = \tan x</math>, given <math>y(0) = 1</math>.

Find the particular solution of the differential

equation <math>(1+\sin x)\frac{dy}{dx} = -x - y\cos x</math>, given <math>y(0) = 1</math>.<br>(Term II. 2021-22C)

1. Find the particular solution of the differential equation <math>x \frac{dy}{dy} + 2y = x^2 \log x</math>, given y(1) = 1.
2. Find the particular solution of the differential equation <math>x \frac{dy}{dx} + y + \frac{1}{1+x^2} = 0</math>, given that <math>y(1) = 0</math>. (Term II, 2021-22)

Find the general solution of the differential equation

<math>x(y^3 + x^3) dy = (2y^4 + 5x^3y)dx</math> (Term II, 2021-22)

Solve the following differential equation:

<math>(y - \sin^2 x)dx + \tan x dy = 0</math> (Term II, 2021-22)

Find the general solution of the differential equation;

<math>(x^3 + y^3)dy = x^2y dx</math> (Term II, 2021-22) (Ap OR Find the general solution of the differential equation

<math>x^2y dx - (x^3 + y^3) dy = 0.</math> (2020)

1. Find the general solution of the differential equation <math>ye^{y} dx = (y^3 + 2x e^{y})dy</math>. (2020)
2. Solve the following differential equation: <math>(1+e^{y/x})dy+e^{y/x}(1-\frac{y}{x})dx=0 (x \neq 0).</math> (2020)
3. Find the particular solution of the differential equation <math>x \frac{dy}{dx} = y - x \tan\left(\frac{y}{x}\right)</math>, given that <math>y = \frac{\pi}{4}</math> at <math>x = 1</math>. (2020) (Ap
4. Find the particular solution of the differential equation <math>\cos y \, dx + (1 + e^{-x}) \sin y \, dy = 0</math> given that <math>y = \frac{\pi}{2}</math> when <math>x = 0</math>. (2020)

Find the general solution of the differential equation

<math>ye^{x/y} dx = (xe^{x/y} + y^2)dy, y \neq 0</math> (2020) An

Solve the differential equation:

<math>xdy-ydx=\sqrt{x^2+y^2}dx</math>, given that <math>y=0</math> when <math>x=1</math>.

(Delhi 2019)