Question

solve x^2y"+5xy'+3y=ln(x)

Answer 1

Answer:

See below.

Explanation:

Assuming that the differential equation reads

x2y''−3xy'+5y=0

The differential equation

x2y''−3xy'+5y=0 is a linear homogeneous differential equation.

Proposing

y=c0xα and substituting

(α(α−4)+5)c0xα=0 and solving for α

α(α−4)+5=(α−2+i)(α−2−i)

then

y=x2(c1xi+c2x−i)

but x=elogex and x±i=e±ilogex

now according to

eiz=cosz+isinz we have the general solution.

y=x2(C1sin(logex)+C2cos(logex))

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