Question

Find the differential equation of the family of curves $y=e^x(A\cos x+B\sin x)$ where $A$ and $B$ are arbitrary constant.

Answer 1

$y=e^x(A\cos x+B\sin x)$

$y^{\prime }=e^x(-A\sin x+B\cos x)+e^x(A\cos x+B\sin x)$

$\Rightarrow y^{\prime \prime }=2e^x(-A\sin x+B\cos x)-2e^x(A\cos x+B\sin x)$

$\therefore y^{\prime \prime }=2(y^{\prime }-y)$