- #1
Xyius
- 508
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Integral simplification with dummy variable "s"
I am looking at the derivation for the solution to solve the partial differential equation of a wave with a time-dependant forcing term h(x,t). In the derivation they get to a point where they need to solve the following second order ODE.
[tex]u''(t)+(\frac{\alpha n \pi}{L})^2u(t)=h(t)[/tex]
The method here would be variation of parameters. I obtained a homogeneous solution of..
[tex]c_1cos(\frac{\alpha n \pi}{L}t)+c_2sin(\frac{\alpha n \pi}{L}t)[/tex]
Which the book also got.
I got this for the particular solution after using variation of parameters...
[tex]\int h(t)cos(\frac{\alpha n \pi}{L}t)dt-\int h(t)sin(\frac{\alpha n \pi}{L}t)dt[/tex]
The book got..
[tex]\frac{L}{n \pi \alpha} \int^{t}_{0} h(s)sin[\frac{\alpha n \pi}{L}(t-s)]ds[/tex]
Now I know these must somehow be the same (unless I messed up using variation of parameters) but I am new to using a dummy variable "s" to simplify integrals like this. I have seen it a couple times but do not have much practice with it. If anyone could help me through this it would be great!
Thanks!
I am looking at the derivation for the solution to solve the partial differential equation of a wave with a time-dependant forcing term h(x,t). In the derivation they get to a point where they need to solve the following second order ODE.
[tex]u''(t)+(\frac{\alpha n \pi}{L})^2u(t)=h(t)[/tex]
The method here would be variation of parameters. I obtained a homogeneous solution of..
[tex]c_1cos(\frac{\alpha n \pi}{L}t)+c_2sin(\frac{\alpha n \pi}{L}t)[/tex]
Which the book also got.
I got this for the particular solution after using variation of parameters...
[tex]\int h(t)cos(\frac{\alpha n \pi}{L}t)dt-\int h(t)sin(\frac{\alpha n \pi}{L}t)dt[/tex]
The book got..
[tex]\frac{L}{n \pi \alpha} \int^{t}_{0} h(s)sin[\frac{\alpha n \pi}{L}(t-s)]ds[/tex]
Now I know these must somehow be the same (unless I messed up using variation of parameters) but I am new to using a dummy variable "s" to simplify integrals like this. I have seen it a couple times but do not have much practice with it. If anyone could help me through this it would be great!
Thanks!