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Question
Show that the differential equation of which \[y = 2\left( x^2 - 1 \right) + c e^{- x^2}\] is a solution is \[\frac{dy}{dx} + 2xy = 4 x^3\]
Solution
We have,
\[y = 2\left( x^2 - 1 \right) + c e^{- x^2}...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = 4x - c e^{- x^2} 2x\]
\[ = 2x\left[ 2 - c e^{- x^2} \right]\]
\[ = - 2x\left[ 2 x^2 - 2 + c e^{- x^2} - 2 x^2 \right]\]
\[ = - 2x\left[ 2\left( x^2 - 1 \right) + c e^{- x^2} - 2 x^2 \right]\]
\[ = - 2x\left[ y - 2 x^2 \right] .............\left[\text{Using }\left( 1 \right) \right]\]
\[ \Rightarrow \frac{dy}{dx} = - 2xy + 4 x^3 \]
\[ \Rightarrow \frac{dy}{dx} + 2xy = 4 x^3\]
Hence, the given function is the solution to the given differential equation.
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