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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
show that the differential equation of which $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is a solution, is $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$
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Answer

The given equation is

$\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}} ...(1)$

Where c is a parameter.

As this equation has one arbitrary constant, we shall get a differential equation of first order.

Differentiating equation (1) with respect to x, we get

$\frac{\text{dy}}{\text{dx}}=2(2\text{x})+\text{ce}^{-\text{x}^{2}}(-2\text{x})$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=4\text{x}-2\text{xce}^{-\text{x}^{2}}$

From (1) and (2), we get

$\frac{\text{dy}}{\text{dx}}=4\text{x}-2\text{x}[\text{y}-2\text{x}^2+2]$

$\Rightarrow\frac{\text{dy}}{\text{dx}}=4\text{x}-2\text{xy}+4\text{x}^3-4\text{x}$

$\Rightarrow\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3$

Hence, $\text{y}=2(\text{x}^2-1)+\text{ce}^{-\text{x}^{2}}$ is the solution to the differential equation $\frac{\text{dy}}{\text{dx}}+2\text{xy}=4\text{x}^3.$

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