Solve the differential equation. x\frac{dy}{dx}-4y=2x^{4}e^{x}

Jason Farmer 2021-09-16 Answered
Solve the differential equation.
xdydx4y=2x4ex
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Expert Answer

berggansS
Answered 2021-09-17 Author has 91 answers
Step 1
According to the given information, it is required to solve the differential equation.
xdydx4y=2x4ex
Step 2
First divide the whole differential equation by x to get the linear differential form.
dydx4yx=2x3ex
Step 3
Now, the general linear differential equation and its solution is:
dydx+P(x)y=Q(x)
solution of differential equation is:
y(x)×IF=Q(x).IFdx+c
where IF=eP(x)dx
Step 4
Now, solve the given using the above definition.
In the given differential equation is:
P(x)=4x,Q(x)=2x3ex
IF=ePdx=e4xdx=e4log(x)=elog(x4)=x4
IF=1x4
the solution of the given differential equation is:
y(1x4)=2x3ex(1x4)dx+c
yx4=2exxdx+c
yx4=2Ei(x)+c(exxdx=Ei(x)+c)
y(x)=2x4Ei(x)+cx4
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