Find at least one first order differential equation from general solution

Question

How can I find at least one first order differential equation when the solution for it is $y = x^4$?

Thanks for your help :)

Answer 1

Most basic equation would be simply $ \frac {dy}{dx}$ = 4$x^3$
with initial condition that y(0)=0

You can get infinite solutions by introducing some constant terms or some specific f(x), and restricting them with specific initial conditions.

For eg. take $$ \frac {dy}{dx} = 4x^3+k$$ (where k is some constant)

So, $$dy=(4x^3+k)dx$$
integrating both sides,
$$y=x^4+kx+c$$
Now, use y(0)=0

How about taking $$ \frac{dy}{dx} = (4x^3 + px^q + r)?$$
$$y=x^4+ \frac {px^{q+1}}{q+1}+rx+c$$
Again, y(0)=0