Solve the ode $W^{\prime }\left( s\right) -2iH\left( s\right) W\left( s\right) -1=0$

Question

Consider the differential equation

$W^{\prime }\left( s\right) -2iH\left( s\right) W\left( s\right) -1=0$, $% s\in I\subset \mathbb{R} $

and consider the functions

$F\left( s\right) =\int\limits_{0}^{s}\sin \left( 2\int\limits_{0}^{u}H\left( t\right) dt\right) du$

$G\left( s\right) =\int\limits_{0}^{s}\cos \left( 2\int\limits_{0}^{u}H\left( t\right) dt\right) du$

where $H:I\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ is functions real of one variable

Show that the general solution to the above differential equation is given by

$W\left( s\right) =\left\{ \left( F\left( s\right) -c_{1}\right) +i\left( G\left( s\right) +c_{2}\right) \right\} \left( F^{\prime }\left( s\right) -iG^{\prime }\left( s\right) \right) $

Can anyone help me solve this differential equation? Well I just know the ode's that I know do not have complex coefficients.

Answer 1

You treat the equation as if it were real. Multiply by $\exp\bigl(-2\,i\int_0^sH(t)\,dt\bigr)$ to get $$ \Bigl(\exp\bigl(-2\,i\int_0^sH(t)\,dt\bigr)\,W\Bigr)'=\exp\bigl(-2\,i\int_0^sH(t)\,dt\bigr). $$ Integrate and remember that $$ \exp\Bigl(-2\,i\int_0^sH(t)\,dt\Bigr)=\cos\Bigl(2\int_0^sH(t)\,dt\Bigr)-\sin\Bigl(2\int_0^sH(t)\,dt\Bigr)i. $$