Question

linear differential equation using integrating factor

Answer 1

Answer:

We multiply both sides of the differential equation by the integrating factor I which is defined as I = e∫ P dx. ⇔ Iy = ∫ IQ dx since d dx (Iy) = I dy dx + IPy by the product rule. As both I and Q are functions involving only x in most of the problems you are likely to meet, ∫ IQ dx can usually be found

Answer 2

Given Question :-

Solve the given linear differential equation

$\rm :\longmapsto\:x\dfrac{dy}{dx} + y = cosx$

$\large\underline{\sf{Solution-}}$

The given Differential equation is

$\rm :\longmapsto\:x\dfrac{dy}{dx} + y = cosx$

$\rm :\longmapsto\:\dfrac{dy}{dx} + \dfrac{y}{x} = \dfrac{cosx}{x}$

On comparing with

$\red{\rm :\longmapsto\:\dfrac{dy}{dx} + py = q \: \: where \: p,q \: \in \: f(x) \: }$

We get

$\red{\rm :\longmapsto\:p = \dfrac{1}{x}}$

and

$\red{\rm :\longmapsto\:q = \dfrac{cosx}{x}}$

To solve this Differential equation, we have to first find Integrating Factor which is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: IF = {e}^{\displaystyle \int \: p \: dx} }}}$

So,

$\rm :\longmapsto\:IF = {e}^{\displaystyle \int \: \frac{1}{x} \: dx}$

$\rm \implies\:IF = {e}^{logx} = x$

Now, Solution of differential equation is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: y \times IF = \displaystyle \int \: (q \times IF) \: dx}}}$

So, solution is

$\rm :\longmapsto\:yx = \displaystyle \int \: x \times \frac{cosx}{x} \: dx$

$\rm :\longmapsto\:yx = \displaystyle \int \: cosx \: dx$

$\rm \implies\:\:\boxed{ \tt{ \: yx = sinx + c \: }}$

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More to Know :-

The other form of linear differential equation is

$\red{\rm :\longmapsto\:\dfrac{dx}{dy} + px = q \: \: where \: p,q \: \in \: f(y) \: }$

To solve this linear differential equation, the following steps have to be followed.

Step 1 : Integrating Factor

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: IF = {e}^{\displaystyle \int \: p \: dy} }}}$

Step 2 : Solution of Linear Differential equation is given by

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: x \times IF = \displaystyle \int \: (q \times IF) \: dy \: }}}$