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/sci/ - Science & Math

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4824970 No.4824970 [Reply] [Original]

Hi,

I've got a question about the approach to finding a particular solution to a second order linear differential equation. As an example:

<span class="math"> \displaystyle y'' - y' = 8(cos(x) + sin(x)) [/spoiler]

My approach was to say that a particular solution has to be of the form

<span class="math">\displaystyle y_p = 8A (B cos(x) + C sin(x)) [/spoiler]

since the function on the right hand side is a zero-grade polynomial multiplied with trigonometric functions.

When I differentiate <span class="math">\displaystyle y_p [/spoiler] and plug it into the equation, I get the term

<span class="math">\displaystyle 0 = 8 cos(x) + 8 sin(x)[/spoiler]

What went wrong?

>> No.4824972

>>4824970
Do you mean <span class="math">y''-y[/spoiler]?
Otherwise you don't have zero on the RHS.

>> No.4824973

>>4824972
Wait, I meant <span class="math">y''+y[/spoiler].

>> No.4824974

Convert it in a system of 2 first order ODEs and plug it in the variation of constant formula.

>> No.4824981

>>4824973
No, it's <span class="math">\displaystyle y'' - y' [/spoiler]. Not sure if I differentiated wrong. Also, the zero would be on the LHS unless you just turned the equation around (but that wouldn't matter, obviously).

>> No.4824987

>>4824981
Then you have error in differentiating. Recheck everything.

>> No.4825018

>>4824987
My bad, I'm stupid. It indeed is <span class="math"> y'' + y'<span class="math">, assuming you're the one who asked.[/spoiler][/spoiler]

>> No.4825040

>My approach was to say that a particular solution has to be of the form
>What went wrong?

This went wrong. I agree it is a reasonable assumption, but if it gets you to a absurd result, it means it is wrong.

Solve it for y' with the variation of constant method.

>> No.4825046

>>4825040
Forgot to mention that you should have realized that this equation is actually a first order equation for y', thus the idea of solving it with the variation of constant method for an order one.

>> No.4825325
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4825325

>variation of constant formula
>variation of constant