Question

Verify that y = − x − 1 is a solution of the differential equation (y − x) dy − (y² − x²) dx = 0.

Answer 1

We have,
\[y = - x - 1...........(1)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = - 1.............(2)\]
Now,
\[\frac{dy}{dx} - \frac{y^2 - x^2}{y - x}\]
\[ = \frac{dy}{dx} - \left( y + x \right)\]
\[ = - 1 - \left( - x - 1 + x \right) ..........\left[ \text{Using }\left( 1 \right) \text{ and }\left( 2 \right) \right]\]
\[ = - 1 + 1 = 0\]
\[ \Rightarrow \frac{dy}{dx} = \frac{y^2 - x^2}{y - x}\]
\[ \Rightarrow \left( y - x \right)dy = \left( y^2 - x^2 \right)dx\]
\[ \Rightarrow \left( y - x \right)dy - \left( y^2 - x^2 \right)dx = 0\]

Hence, the given function is the solution to the given differential equation.