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Linear Differential Equation - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Linear Differential Equation is considered one the most difficult concept.
85 Questions around this concept.

Solve by difficulty

The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y(1)=1$ is

A

$y=x\; \ln \; x+x$

B

$y= \ln \; x+x$

C

$y=x\; \ln \; x+x^{2}$

D

$y=x\, e^{(x-1)}$

Let y(x) be the solution of the differential equation

$(x\, \log \, x)\frac{dy}{dx}+y=2x\, \log x,\left ( x\geq 1 \right ).$ Then y(e) is equal to :

A

e

B

0

C

2

D

2e

Concepts Covered - 1

Linear Differential Equation

The linear differential equations are those in which the variable and its derivative occur only in the first degree.

An equation of the form

$\frac{dy}{dx}+P(x)\cdot y=Q(x)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\text{(i)}$

Where P(x) and Q(x) are functions of x only or constant is called a linear equation of the first order.

To solve the differential equation (i)
$\\\text{multiply both sides of Eq (i) by }\int e^{P(x)\;dx}\text{, we get}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;}\mathrm{e}^{\int \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{P}(\mathrm{x}) \mathrm{y}\right)=\mathrm{e}^{\int \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}} \cdot \mathrm{Q}(\mathrm{x})\\\\\mathrm{i.e.\;\;\;\;\;\;\;\;e^{\int P(x) d x} \cdot \frac{d y}{d x}+y .P(x)\frac{d}{d x}\left(e^{\int P(x) d x}\right)=Q e^{\int P(x) d x}}\\\\\mathrm{or\;\;\;\;\;\;\;\;\;\frac{d}{d x}\left(y e^{\int P(x) d x}\right)=e^{\int P(x) d x} \cdot Q(x)}\\\\\text{Integrating both sides, we get}\\\\\mathrm{or\;\;\;\;\;\;\;\;\;\int{d}\left(y e^{\int P(x) d x}\right)=\int\left (e^{\int P(x) d x} \cdot Q(x) \right )dx}\\\\\mathrm{\Rightarrow \;\;\;\;\;\;\;\;\;\mathrm{y} \mathrm{e}^{\int \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}}=\int \mathrm{Q}(\mathrm{x}) \mathrm{e}^{\int \mathrm{P}(\mathrm{x}) \mathrm{d} \mathrm{x}} \mathrm{d} \mathrm{x}+\mathrm{C}}$

Which is the required solution of the given differential equation.

$\\\mathrm{The\;term\; e^{\int P(x)\;dx}\;\;which \;convert\;the\;left\;hand\;expression\;of\;the\;equation}\\\text{into a perfect differential is called an Integrating factor (IF).}\\\\ {\text { Thus, we remember the solution of the above equation as }} \\ {\qquad y(\mathrm{IF})=\int Q(\mathrm{IF}) d x+C}$

NOTE :

Sometimes a given differential equation becomes linear if we take x as the dependent variable and y as the independent variable.

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