ODE:
$$X''(t)+A(t)X'(t)+B(t)X(t)=F(t),$$
IC's: $$X(0)=U_1, $$ $$X'(0)=U_2$$
where
$X(t)=[X_1 X_2...X_n]^T$ and $U_1, U_2,F(t)$ are $n\times1$ matrices,
$A(t), B(t)$ are $n\times n$ matrices.
Question: How to solve the 2nd order ODE with variable coefficients by Legendre Wavelets?
(I know to solve 2nd order ODE with constant coefficients by Legendre Wavelets. But I can' t solve the ODE with variable coefficients)
My try: Let $X''(t)=C^T \Psi(t).$
Then we take integral from $0$ to $t$, we have
$X'(t)=C^T P \Psi(t)+X'(0)=C^T P \Psi(t)+\bar{C}_0^T \Psi(t)$
where $\int_{0}^{t}C^T \Psi(\hat{t})d\hat{t}=C^T P \Psi(t)$ and $X'(0)=\bar{C}_0^T \Psi(t).$
and similarly we take integral from $0$ to $t$, we have
$X(t)=C^T P_2 \Psi(t)+\bar{C}_0^T \Psi(t)+C_0^T \Psi(t)$ where $X(0)=C_0^T \Psi(t).$
Let $A(t)=\bar{A}^T\Psi(t) $ and $B(t)=\bar{B}^T\Psi(t) $ and $F(t)=\bar{F}^{T}\Psi(t). $
Substituting to all above equations to ODE, we have $$C^T \Psi(t)+\bar{A}^T\Psi(t)\big(C^T P \Psi(t)+\bar{C}_0^T \Psi(t)\big)+\bar{B}^T \Psi(t)\big(C^T P_2 \Psi(t)+\bar{C}_0^T \Psi(t)+C_0^T \Psi(t)\big)=\bar{F}^{T}\Psi(t) $$
Is it right? And then?