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