Question

Form a differential equation representing the given family
of curves by eliminating arbitrary constant a and b.

$y=e^x\left(a\cos x+b\sin x\right)$

Answer 1

$y=e^x(acosx+bsinx)$

$y^{{}^{\prime }}=e^x(acosx+bsinx)+e^x(-asinx+bcosx)$

$y^{{}^{\prime }}=y+e^x(-asinx+bcosx)$

$y^{{}^{\prime \prime }}=y^{{}^{\prime }}+e^x(-asinx+bcosx)-e^x(acosx+bsinx)$

$y^{{}^{\prime \prime }}=y^{{}^{\prime }}+y^{{}^{\prime }}-y-y$

$y^{{}^{\prime \prime }}=2y{}^{\prime }-2y$ is the required differential equation.