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Differential equations with variables separable - Practice Questions & MCQ

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

Quick Facts

  • 32 Questions around this concept.

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QuestionEASY

 If \left ( 2+\sin x \right )\: \frac{dy}{dx}\: +\left ( y+1 \right )\cos x= 0

and y(0)=1,then  y\left ( \frac{\pi }{2} \right )   is equal to :

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If    \dpi{100} \frac{dy}{dx}=y+3> 0\; and\; y(0)=2,then\; y(1n\: 2)  is equal to

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Concepts Covered - 2

Differential equations with variables separable

\\\mathrm{The \;differential\;of\;the\\;form\;\frac{dy}{dx}=f(x)g(y)\;where\;f(x)\;is\;function\;of\;x}\\\mathrm{and\;g(y)\;is\;a\;function\;of\;y,\;are\;said\;to\;be\;variable\;separrable\;form.}

Rewrite the equation as

\\\mathrm{\frac{dy}{g(y)}=f(x)dx\;\;\;\;\;\;\;\;[where\;g(y)\neq0]}\\\\\text{This\;process is separating the variables.}\\\mathrm{Now,\; integrate\;both\;side,\;we\;get}\\\\\mathrm{\int \frac{dy}{g(y)}=\int f(x)\;dx+c}

By this, we get the solution of differential equation

 

Let’s see one illustration for better understanding

Solution of the differential equation \\\mathrm{\frac{dy}{dx}=(e^x+1)(y^2+1)}

Rewrite the differential equation as

\\ \frac{\mathrm{dy}}{1+\mathrm{y}^{2}}=\left(\mathrm{e}^{\mathrm{x}}+1\right) \mathrm{d} \mathrm{x}\\\\\text{Integrating both sides, we get }\\\\ \int\frac{\mathrm{dy}}{1+\mathrm{y}^{2}}=\int \left(\mathrm{e}^{\mathrm{x}}+1\right) \mathrm{d} \mathrm{x} \\\Rightarrow \tan ^{-1} y=e^{x}+x+c\\\Rightarrow y=\tan \left(e^{x}+x+c\right)

Differential Equation Reducible to Variable Separable Form

A differential equation of the form \\\mathrm{\frac{dy}{dx}=f(ax+by+c)} where a, b and c are constants, can be converted into an equation with variables separable by the substitution v = ax + by + c.

\\\mathrm{\frac{dy}{dx}=f(ax+by+c)\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots(i)}\\\\\mathrm{v = ax + by + c.}\\\therefore \quad \frac{d v}{d x}=a+b \frac{d y}{d x} \text { or, } \frac{d y}{d x}=\frac{\frac{d v}{d x}-a}{b}\\\\\Rightarrow \;\;\frac{\frac{\mathrm{d} v}{\mathrm{dx}}-\mathrm{a}}{\mathrm{b}}=\mathrm{f}(\mathrm{v}) \Rightarrow \frac{\mathrm{d} \mathrm{v}}{\mathrm{dx}}=\mathrm{bf}(\mathrm{v})+\mathrm{a}\\\\\Rightarrow \;\;\;\frac{\mathrm{d} \mathrm{v}}{\mathrm{bf}(\mathrm{v})+\mathrm{a}}=\mathrm{dx}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\mathrm{(ii)}\\\\\text{In the differential equation (ii), the variables x and v are separated.}\\\text{Integrating (ii), we get}\\\Rightarrow \;\;\;\;\int \frac{\mathrm{dv}}{\mathrm{bf}(\mathrm{v})+\mathrm{a}}=\int \mathrm{dx}+\mathrm{C}\\\\\Rightarrow \;\;\;\int \frac{\mathrm{d} \mathrm{v}}{\mathrm{bf}(\mathrm{v})+\mathrm{a}}=\mathrm{x}+\mathrm{C}, \text { where } \mathrm{v}=\mathrm{ax}+\mathrm{by}+\mathrm{c}

This represents the general solution of the differential equation (i).

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Differential equations with variables separable
Differential Equation Reducible to Variable Separable Form

Study other Related Concepts

Differential equations with variables separable Current Topic

Differential Equation
Formation of Differential Equation and Solutions of a Differential Equation
Differential equations with variables separable
Homogeneous Differential Equation
Linear Differential Equation
Bernoulli’s Equation
Orthogonal Trajectory
Exact Differential Equation
Application of Differential Equation

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Books

Reference Books

Differential equations with variables separable

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 10.4

Line : 45

Differential Equation Reducible to Variable Separable Form

Mathematics for Joint Entrance Examination JEE (Advanced) : Calculus

Page No. : 10.4

Line : 47

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