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Differential Equations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDifferential Equations2 Marks
Question
Verify that the function $x y=a e^{x}+b e^{-x}+x^{2} ($implicit or explicit$)$ is a solution of the differential equation $x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2=0$
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Answer

It is given that $xy = a e^x + b e^{-x} + x^2$
Now, differentiating both sides $w.r.t. x,$ we get,
$\Rightarrow y+x \cdot \frac{d y}{d x}=a \cdot e^{x}-b e^{-x}+2 x ...(i)$
Now, Again differentiating both sides $w.r.t. x$, we get,
$\Rightarrow \frac{d y}{d x}+\frac{d y}{d x}+x \cdot \frac{d^{2} y}{d x^{2}}=a e^{x}+b e^{-x}+2 ...(ii)$
Now, Using Equations. $(i)$ and $(ii),$ we get,
$\text{LHS}\ x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2$
$= a e^{x}+b e^{-x}+2-\left[a e^{x}+b e^{-x}+x^{2}\right]+x^{2}-2$
$= a e^{x}+b e^{x}+2 -a e^{x}-b e^{-x}-x^{2}+x^{2}-2 $
$= 0$
$\Rightarrow \text{LHS = RHS}.$
Therefore, the given function is the solution of the corresponding differential equation.

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