Question

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

$y=e^{2x}\left(a+bx\right)$

Answer 1

$y=e^{2x}(a+bx)$

Differentiating we get,

$\frac{dy}{dx}=2e^{2x}(a+bx)+b{e}^{2x}$

$y^{{}^{\prime }}=2y+b{e}^{2x}$

$⟹(y^{{}^{\prime }}-2y)e^{-2x}=b$

Differentiating again, we get

$y^{{}^{\prime \prime }}=2y^{{}^{\prime }}+2b{e}^{2x}$

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

$y^{{}^{\prime \prime }}=4y^{{}^{\prime }}-4y$