Question

Verify that y = cx + 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,
\[y = cx + 2 c^2..............(1)\]
Differentiating both sides of (1) with respect to x, we get

\[\frac{dy}{dx} = c...........(2)\]
Now,
\[2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y\]
\[ = 2 c^2 + cx - cx - 2 c^2 = 0 ...........\left[\text{Using }\left( 1 \right)\text{ and }\left( 2 \right) \right]\]
\[ \Rightarrow 2 \left( \frac{dy}{dx} \right)^2 + x\frac{dy}{dx} - y = 0\]
Hence, the given function is the solution to the given differential equation.