Solve differential equation (x+2)y′+4y=(3x + 6)^-2 lnx

waigaK 2021-02-24 Answered

Solve differential equation (x+2)y+4y=(3x+6)2lnx

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Expert Answer

ottcomn
Answered 2021-02-25 Author has 97 answers

((x+2)y+4y)/((x+2))=((3x+6)2lnx)/((x+2))
y+(4/(x+2))y=lnx/(9(x+2)3) (1)
Now, we know that solution of differential equation(y+p(x)y=q(x)) is given by
y(I.F.)=q(x)(I.F.)dx (2)
where I.F.=ep(x)dx
So, finding integrating factor of equation (1)
I.F.=e4/(x+2)dx
=e4dx/(x+2) (:dx/(x+a)=ln(x+a)+c)
=e4(ln(x+2))
=eln(x+2)4
=(x+2)4
So, solution of given differential equation will be
y(x+2)4=lnx/(9(x+2)3)(x+2)4dx
y(x+2)4=1/9(x+2)lnxdx
y(x+2)4=1/9[lnx(x+2)dx1/x[(x+2)dx]dx]
=1/9[((x+2)2lnx)/21/x((x+2)2)/2dx]
=((x+2)2lnx)/181/2(x2+4x+4)/xdx
=((x+2)2lnx)/181/2(x2/2+4x+4lnx)+c

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