Question

Show that y = e^−x + ax + b is solution of the differential equation\[e^x \frac{d^2 y}{d x^2} = 1\]

 

Answer 1

We have,
\[y = e^{- x} + ax + b.............(1)\]
Differentiating both sides of equation (1) with respect to `x`, we have

\[\frac{dy}{dx} = - e^{- x} + a..............(2)\]

Differentiating both sides of equation (2) with respect to `x`, we have

\[\frac{d^2 y}{d x^2} = e^{- x} \]
\[ \Rightarrow e^x \frac{d^2 y}{d x^2} = 1\]

Hence, the given function is a solution of the given differential equation.