y=e2x(a+bx)
It consists of 2 arbitrary constants.
This means we can differentiate it twice to get the differential equation.
Multiply by e−2x on both sides, we get
e−2xy=a+bx
Differentiating wrt x, we get
e−2xdydx+(−2)e−2xy=b⇒e−2x(dydx−2y)=b
Again, differentiating wrt x, we get
e−2x(d2ydx2−2 dydx)+(−2)e−2x(dydx−2y)=0⇒e−2x(d2ydx2−4dydx+4y)=0⇒d2ydx2−4dydx+4y=0 [∵e−2x≠0]
This is the required differential equation.