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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations2 Marks
Question
Verify that the function $y=x \sin 3 x$ (implicit or explicit) is a solution of the differential equation $\frac{d^{2} y}{d x^{2}}+9 y-6 \cos 3 x=0$ 
✓

Answer

It is given that y = x sin 3x
Now, differentiating both sides w.r.t. x , we get,
$\frac{d y}{d x}=\frac{d}{d x}(x \sin 3 x)=\sin 3 x+x . \cos 3 x .3$ 
$\Rightarrow \frac{d y}{d x}=\sin 3 x+3 x \cos 3 x$ 
Now, again differentiating both sides w.r.t. x, we get,
$\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}(\sin 3 x)+3 \frac{d}{d x}(x \cos 3 x)$ 
$\Rightarrow \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=3 \mathrm{cos} 3 \mathrm{x}+3[\cos 3 \mathrm{x}+\mathrm{x}(-\sin 3 \mathrm{x}) \cdot 3]$ 
$\Rightarrow \frac{d^{2} y}{d x^{2}}=6 \cos 3 x-9 x \sin 3 x$ 
Now, substituting the value of $\frac{d^{2} y}{d x^{2}}$ in the LHS of the given differential equation, we get,
$\frac{d^{2} y}{d x^{2}}+9 y-6 \cos 3 x$ 
= (6.cos 3x - 9x sin 3x) + 9x sin 3x - 6 cos 3x
= 0 = RHS
Therefore, the given function is the solution of the corresponding differential equation.

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