Variation of parameters method for differential equations. Change the variable x=e^{t} and

Caitlyn Cole

Caitlyn Cole

Answered question

2022-04-21

Variation of parameters method for differential equations.
Change the variable x=et and then find the general solution for the following differential equation 2x2y6xy+8y=2x+2x2lnx
It seems a little suspicious that i can factor out 2 and x first. x2y3xy+4y=x(ln(x)+1) and if we factor out x2 now we end up with y3xy+4x2y=xlnx+1x
(1) Therefore substituting x=et (1) now becomes y3ety+4e2ty=e+1et. We need constant coefficients in order to use the variation of parameters method am i right?

Answer & Explanation

Norah Small

Norah Small

Beginner2022-04-22Added 12 answers

Answer:
After change the variable x=et we get
y4y+4y=te2t+et
Then y=t3e2t+6et6+(C1+C2t)e2t=
=x2ln3x+6x6+(C1+C2lnx)x2

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