Question

what is the differential equation having a solution y=cx²+cx?

Answer 1

Consider the solution $y=cx+cx^2$

Require to find the differential equation from the given solution.

To find the differential equation, let us eliminate the parameter $c$ by differentiating the given equation with respect to $x$

Differentiating the given equation $y=cx+cx^2$ with respect to x , we get

$y'=c(1)+c(2x)$

Implies, we get $y'=c+2cx$

Implies, we get $y'=c(1+2x)$

$\Rightarrow c=\frac{y'}{1+2x}$

Substituting the value of $c$ in the given equation $y=cx+cx^2$ , we get

$\Rightarrow y=x[\frac{y'}{1+2x}]+x^2[\frac{y'}{1+2x}]$

$\Rightarrow y(1+2x)=xy'+x^2y'$

$\Rightarrow y(1+2x)=y'(x+x^2)$

Therefore, the required differential equation is

$y'=\frac{y(1+2x)}{(x+x^2)}$