DE I don't get

  • Thread starter hola
  • Start date
  • #1
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1. Use substitution x = e^t to transfer equation into one with constant coefficients and solve:
x^2 y'' -3xy' + 13y = 4 + 3x
My work:
Okay, we have
e^2t y'' - 3e^t y' + 13y = 4 + 3e^t.

Now I am absolutely stuck. No idea how to solve it. Help!
 

Answers and Replies

  • #2
HallsofIvy
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You didn't complete the substitution! Your y' and y" are still with respect to x, not t.

Use the chain rule: dy/dx= dy/dt dt/dx. Since x= et, t= ln x and
dt/dx= 1/x. That is, dy/dx= (1/x)(dy/dt).
d2y/dx2= d((1/x)(dy/dt)/dx= (-1/x2)dy/dt+ (1/x)d(dy/dt)/dx= ((-1/x2)dy/dt+(1/x)(1/x)d2y/dt2=.
Now, x2 y'' = d2y/dt2- dy/dt
and -3xy'= -3 dy/dt
so the differential equation is
d2y/dt2- 4dy/dt+ 13y= 4+ 3ett

That should be easy to solve!
 

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