Ordinary Differential Equations

  • Thread starter NeoDevin
  • Start date
  • #1
299
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Homework Statement


Use variation of parameters to find the general solution of:

[tex] x^2 y'' + xy' + 9y = -tan(3ln(x)) [/tex]


Homework Equations



[tex]y_p = y_2 (x) \int \frac{y_1 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx - y_1 (x) \int \frac{y_2 (x) g(x)}{W[y_1 (x),y_2 (x)]} dx [/tex]

The Attempt at a Solution



Solution for the homogeneous Euler equation is given by
[tex]
y_1 (x) = cos(3ln(x)), y_2 (x) = sin(3ln(x))
[/tex]

I try using these solutions in the equation for variation of parameters with:
[tex] W[y_1 , y_2] = \frac{3}{x} [/tex]
[tex] g(x) = -tan(3ln(x)) [/tex]

The first term I can evaluate, but the second term gives me:
[tex] y_1 (x) \int \frac{x sin^2 (3ln(x))}{3 cos(3ln(x))} dx [/tex]

which doesn't seem solvable by any methods I know...

Any suggestions on what to do from here, or where I went wrong on the way would be helpful.
 

Answers and Replies

  • #2
150
0
You have to set it up as y" +..... to use your variation of parameters formula
 
  • #3
299
1
so the solutions for the homogeneous equation are the same, right? but I need to divide through by [itex] x^2 [/itex] before I apply the formula? so
g(x) becomes:

[tex] g(x) = -\frac{tan(3ln(x))}{x^2} [/tex]
 

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