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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
$\text{y}=\text{x}\sin​3\text{x}​$ : $\frac{\text{d}^2\text{y}}{\text{dx}^2}+9\text{y}-6\cos3\text{x}=0$
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Answer

The given differential equation is

$\text{y}=\text{x}\sin3\text{x}​\ \ ...(1)$

$\Rightarrow\ \ \frac{\text{dy}}{\text{dx}}=\text{x}\cos3\text{x}. 3+\sin3\text{x}$

$\text{and}\ \ \frac{\text{d}^2\text{y}}{\text{dx}^2}=3\{\text{x}(-\sin3\text{x}. 3)+\cos3\text{x}\}+\cos3\text{x}. 3$

$\text{or}\ \ \frac{\text{d}^2\text{y}}{\text{dx}^2}=-9\text{x}\sin3\text{x}+6\cos3\text{x}$

$\text{or}\ \ \frac{\text{d}^2\text{y}}{\text{dx}^2}=-9\text{y}+6\cos3\text{x}\ \ [\because​​​​​​\text{of (1)}]$

$\text{or}\ \ \frac{\text{d}^2\text{y}}{\text{dx}^2}=-9\text{y}+6\cos3\text{x}=0.$

Hence the result.

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