# ODEPR1

For the equation $u''+(\delta +\epsilon \cos t+\epsilon \sin 2t)u=0\,$ show that the conditions for parametric resonance are $\epsilon =0\,$ and $\delta =k^{2}\,$ or $(k+1/2)^{2},(k=0,1,2,...)\,$.

Let $\epsilon =0\,$.

Then $u''+\delta u=0\,$ has two L.I. solutions $\cos {\sqrt {\delta }}t\,$ and ${\frac {\sin {\sqrt {\delta }}t}{{\sqrt {\delta }}}}\,$.

So the fundamental matrix is $X(t)={\begin{bmatrix}\cos {\sqrt {\delta }}t&{\frac {\sin {\sqrt {\delta }}t}{{\sqrt {\delta }}}}\\-{\sqrt {\delta }}\sin {\sqrt {\delta }}t&\cos {\sqrt {\delta }}t\end{bmatrix}}\,$

Notice $X(0)=I\,$ (the identity matrix) so this is a principal fundamental matrix.

The monodromy matrix $B\,$ is the inverse of the fundamental matrix evaluated at zero times the fundamental matrix evaluted at the period of the coefficients.

$B=X^{{-1}}(0)\cdot X(2\pi )=I\cdot X(2\pi )=X(2\pi )={\begin{bmatrix}\cos {\sqrt {\delta }}2\pi &{\frac {\sin {\sqrt {\delta }}2\pi }{{\sqrt {\delta }}}}\\-{\sqrt {\delta }}\sin {\sqrt {\delta }}t&\cos {\sqrt {\delta }}2\pi \end{bmatrix}}\,$

The eigenvalues of $B\,$ are $\rho _{{1,2}}=\phi \pm {\sqrt {\phi ^{2}-1}}\,$.

Also it is true that $\phi ={\frac {1}{2}}{\mathrm {Trace}}(B)=\cos {\sqrt {\delta }}2\pi \,$.

So now $\rho _{{1,2}}=\cos {\sqrt {\delta }}2\pi \pm i\sin {\sqrt {\delta }}2\pi \,$.

It is true that $-1\leq \phi \leq 1\,$.

In case $-1<\phi <1\,$, the solutions are bounded, oscillatory, and stable.

In case $\phi =-1,\,\,\,\cos {\sqrt {\delta }}2\pi =-1\implies \delta =\left(k+{\frac {1}{2}}\right)^{2}\,$.

In case $\phi =1,\,\,\,\cos {\sqrt {\delta }}2\pi =1\implies \delta =k^{2}\,$.