How to prove this theorem? I have a doubt.
Theroem: Suppose that $w(x,\alpha)$ solved $\ddot{w}+\frac{2}{x}\dot{w}+\alpha^{2}w=0$ with $w(0)=1$, $\dot{w}(0)=0$, $w(1)=0$. Then $v(x,\alpha) := \omega w(x,\alpha)$ solves $\ddot{v}+\frac{2}{x}\dot{v}+\Theta^{2}v^{n}=0$ with $v(0)=\omega$, $\dot{v}(0)=0$, $v(1)=0$ provided that \begin{equation} \omega = \left(\frac{\alpha^{2}}{\Theta^{2}}\right)^{1/(n-1)}. \end{equation}
Demonstration:
- \begin{equation} v(0) = \omega w(0) = \omega 1 = \omega\end{equation}
- \begin{equation} \dot{v}(0) = \omega \dot{w}(0) = \omega 0 = 0\end{equation}
- \begin{equation} v(1) = \omega w(1) = \omega 0 = 0\end{equation}
- Substituting $v=\omega w$ into $\ddot{v}+\frac{2}{x}\dot{v}+\Theta^{2}v^{n}=0$ and cancelling common factors, we obtain $\ddot{w}+\frac{2}{x}\dot{w}+\alpha^{2}w^{n}=0$.
My question is: $\ddot{w}+\frac{2}{x}\dot{w}+\alpha^{2}w=0$ then $\ddot{w}+\frac{2}{x}\dot{w}+\alpha^{2}w^{n}=0$? I guess not. I can´t demostrate this last part of the theorem.
Referene: Boyd, J. P. (2011). Chebyshev spectral methods and the Lane-Emden problem. Numerical Mathematics: Theory, Methods and Applications, 4(2), 142-157.