Find the general solution of the given differential equation. x^2y'+x(x+7)y=e^x y=

PEEWSRIGWETRYqx

PEEWSRIGWETRYqx

Answered question

2021-12-21

Find the general solution of the given differential equation.
x2y+x(x+7)y=ex
y=

Answer & Explanation

enlacamig

enlacamig

Beginner2021-12-22Added 30 answers

The differential equation is an example.
x2y+x(x+7)y=ex
On rewriting it, we get
y+(1+2x)y=exx2...(1)
Standard form of the linear differential equation dydx+P(x)y=f(x)
Clearly equation (1) is in the standard form
Compare the differential equation (1) with standard orm, and identity
P(x)=2+2x and f(x)=exx2
The dunctions P(x)=2+2x and f(x)=exx2 are continuous on (0,)
The integrating facrote is
eP(x) dx =e(1+2x) dx 
=ex+2lnx
exelnx2
=x2ex
The standard form multiplied by the integrating factor
x2ex(y+(1+2x)y)=x2exexx2
x2exy+x2exy+2xexy=e2x
x2ex dy  dx +x2yd dx (ex)+exyd dx (x2)=e2x
d dx (x2exy)=e2x
x2exy=e2x dx 
x2exy=12e2x+c
x2exy=12x2ex+cx2ex
Hence the general solution of the given differential equation is
 

ol3i4c5s4hr

ol3i4c5s4hr

Beginner2021-12-23Added 48 answers

answer Part 1 12x2ex+Cx2ex
Part 2 (0,)
Part 3 Cx2ex
RizerMix

RizerMix

Expert2021-12-29Added 656 answers

Thank you!!!!!!!

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