Question

Show that y = C x + 2C² is a solution of the differential equation \[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0.\]

Answer 1

We have,

\[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0 . . . . . \left( 1 \right)\]

Now,

y = C x + 2C²

\[\Rightarrow\frac{dy}{dx}=C\]

\[\text{Putting }\frac{dy}{dx} = C\text{ and }y = Cx + 2 C^2\text{ in (1), we get}\]

\[\text{LHS }= 2 \left( C \right)^2 + x\left( C \right) - \left( Cx + 2 C^2 \right)\]

\[ = 2 C^2 + xC - xC - 2 C^2 \]

\[ = 0 =\text{ RHS}\]

Thus, y = C x + 2C² is the solution of the given differential equation.