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How to calculate this linear differential equation
i've been struggling with this for quite some time now. It's fairly basic but just can't seem to get past it. Help is much appreciated.
P(x) = Ae^(kx) + Be^(-kx) with some constants A, B, and k.
Y(x) = -k*P(x) = -kAe^(kx) + -kB*e^(-kx)
I ended up with this integral for solving it, but it's not useful.
A:
\begin{align} Y(x) &= -kAe^{kx}-kB*e^{ -kx} \\
Y'(x) &= kAe^{kx}-kB*e^{ -kx} \\
Y''(x) &= k^2Ae^{kx}-kB*e^{ -kx}+kAe^{kx}-kB*e^{ -kx} \\
&= k^2Ae^{kx}+kB*(e^{ -kx}+e^{kx})
\end{align}
So the solution is
\begin{align} \pmatrix{Y \\ Y' \\ Y''} &= \pmatrix{Ae^{kx} \\ -kB*e^{ -kx} \\ k^2Ae^{kx}+kB*(e^{ -kx}+e^{kx})} \\ &= \pmatrix{Ae^{kx} \\ -kB*e^{ -kx} \\ kAe^{kx}+k^2Ae^{kx}+kB*(e^{ -kx}+e^{kx})} \\
&= \pmatrix{Ae^{kx}+kB*(e^{kx}-e^{ -kx}) \\ -kB*e^{ -kx} \\ kAe^{kx}+k^2Ae^{kx}+ 0b46394aab
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