Solve the differential equation.

$x\frac{dy}{dx}-4y=2{x}^{4}{e}^{x}$

Jason Farmer
2021-09-16
Answered

Solve the differential equation.

$x\frac{dy}{dx}-4y=2{x}^{4}{e}^{x}$

You can still ask an expert for help

berggansS

Answered 2021-09-17
Author has **91** answers

Step 1

According to the given information, it is required to solve the differential equation.

$x\frac{dy}{dx}-4y=2{x}^{4}{e}^{x}$

Step 2

First divide the whole differential equation by x to get the linear differential form.

$\frac{dy}{dx}-\frac{4y}{x}=2{x}^{3}{e}^{x}$

Step 3

Now, the general linear differential equation and its solution is:

$\frac{dy}{dx}+P\left(x\right)y=Q\left(x\right)$

solution of differential equation is:

$y\left(x\right)\times IF=\int Q\left(x\right).IFdx+c$

where$IF={e}^{\int P\left(x\right)dx}$

Step 4

Now, solve the given using the above definition.

In the given differential equation is:

$P\left(x\right)=\frac{-4}{x},Q\left(x\right)=2{x}^{3}{e}^{x}$

$IF={e}^{Pdx}={e}^{\int -\frac{4}{x}dx}={e}^{-4\mathrm{log}\left(x\right)}={e}^{\mathrm{log}\left({x}^{-4}\right)}={x}^{-4}$

$IF=\frac{1}{{x}^{4}}$

the solution of the given differential equation is:

$y\left(\frac{1}{{x}^{4}}\right)=\int 2{x}^{3}{e}^{x}\left(\frac{1}{{x}^{4}}\right)dx+c$

$\frac{y}{{x}^{4}}=2\int \frac{{e}^{x}}{x}dx+c$

$\frac{y}{{x}^{4}}=2{E}_{i}\left(x\right)+c(\int \frac{{e}^{x}}{x}dx={E}_{i}\left(x\right)+c)$

$y\left(x\right)=2{x}^{4}{E}_{i}\left(x\right)+c{x}^{4}$

According to the given information, it is required to solve the differential equation.

Step 2

First divide the whole differential equation by x to get the linear differential form.

Step 3

Now, the general linear differential equation and its solution is:

solution of differential equation is:

where

Step 4

Now, solve the given using the above definition.

In the given differential equation is:

the solution of the given differential equation is:

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