For each of the exercises given below, verify that the given func… - Vidyadip

Question

For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.

$\text{y}=\text{x}\sin3\text{x}$

:

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

✓

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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