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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations5 Marks
Question
verify that $\text{y}=\text{-x}-1$ is a solution of the differential equation $(\text{y}-\text{x})\text{dy}-(\text{y}^2-\text{x}^2)\text{dx}=0.$
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Answer

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

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