Please find attached a snap. Can anyone show me the steps of getting equation 3.15 solved to equation 3.17?enter image description here
Hyperbolic function in Second Order DE
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ordinary-differential-equations
1 Answers
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Your ODE is: $$y''=\xi^2y.$$ You can use Eulers exponential ansatz $y=\exp(\lambda x)$ plug this into the ODE to get:
$$(\lambda^2-\xi^2)\exp(\lambda x)=0.$$
This can only be true if $\lambda^2=\xi^2$, as $\exp(\lambda x)\neq 0$. So we obtain $\lambda_{1/2}=\pm \xi$. We found two linearly independent solutions for the ODE. In order to construct the general solution we use linear combinations of these solutions:
$y(x)=c_1\exp(\xi x)+c_2\exp(-\xi x).$
Use the boundary condtions to determine $c_1$ and $c_2$. Can you complete it from here?
If you rewrite $c_1=a_1/2+b_1/2$ and $c_2 = a_1/2-b_1/2$ you can obtain: $$y(x)=a_1(\exp(\xi x)+\exp(-\xi x))/2+b_1(\exp(\xi x)-\exp(-\xi x))/2.$$
The expressions in the bracket are nothing but $\cosh(\xi x)$ and $\sinh(\xi x)$.