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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations3 Marks
Question
verify that $\text{y}=\text{cx}+2\text{c}^2$ is a solution of the differential equation $2\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2-\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=0$
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Answer

We have,
$\text{y}=\text{cx}+2\text{c}^2\ ...(1)$
Differentiating both sides of (1) with respect to x, we get
$\frac{\text{dy}}{\text{dx}}=\text{c}\ ...(2)$
Now,
$2\Big(\frac{\text{dy}}{\text{dx}}\Big)^2+\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}$
$=2\text{c}^2+\text{cx}-\text{cx}-2\text{c}^2=0$
$\Rightarrow\Big(\frac{\text{d}\text{y}}{\text{dx}}\Big)^2+\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=0$
Hence, the given is the solution to the given differential equation.

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