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Differential Equations — Maths STD 12 Science — Question

Gujarat 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 ex + b e-x + x2
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,
LHS  $x \frac{d^{2} y}{d x^{2}}+2 \frac{d y}{d x}-x y+x^{2}-2$ 
= $a e^{x}+b e^{-x}+2-\left[a e^{x}+b e^{-x}+x^{2}\right]+x^{2}-2$
= $\begin{aligned} &a e^{x}+b e^{x}+2 -a e^{x}-b e^{-x}-x^{2}+x^{2}-2 \end{aligned}$
= 0
$\Rightarrow$ LHS = RHS.
Therefore, the given function is the solution of the corresponding differential equation.

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