Verify that`y=c x+2c^2` is a solution of the differential equation `2((dy)/(dx))^2+x(dy)/(dx)-y=0.`

Explanation

Given: `y=c x+2 c^{2}` ……(1)


Differentiating equation(1) with respect to x,


`frac{d y}{d x}=c` ……..(2)


`2(frac{d y}{d x})^{2}+x frac{d y}{d x}-y`


`Rightarrow2 c^{2}+c x-c x-2 c^{2}=0` ……[Use value of equation(1) and (2)]


`Rightarrow 2(frac{d y}{d x})^{2}+x frac{d y}{d x}-y=0`


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