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

PEEWSRIGWETRYqx 2021-12-21 Answered
Find the general solution of the given differential equation.
x2y+x(x+7)y=ex
y=
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Expert Answer

enlacamig
Answered 2021-12-22 Author has 30 answers

Consider the differential equation
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
Multiply the standart form, with the integrating factor
x2ex(y+(1+2x)y)=x2exexx2
x2exy+x2exy+2xexy=e2x
x2exdydx+x2yddx(ex)+exyddx(x2)=e2x
ddx(x2exy)=e2x
x2exy=e2xdx
x2exy=12e2x+c
x2exy=12x2ex+cx2ex
Hence the general solution of the given differential equation is

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ol3i4c5s4hr
Answered 2021-12-23 Author has 48 answers
answer Part 1 12x2ex+Cx2ex
Part 2 (0,)
Part 3 Cx2ex
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RizerMix
Answered 2021-12-29 Author has 438 answers

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