TITLE: What is the antiderivative of $\left(\frac{dy}{dx}\right)^3$
QUESTION [5 upvotes]: I have the following differential equation: 
$$\frac{dy}{dx} + \left(\frac{dy}{dx}\right)^3 = C$$
How would one solve this? I tried integrating both sides and I got: 
$$y + \int\left(\frac{dy}{dx}\right)^3\;dx = Cx + D$$
What is $$\int\left(\frac{dy}{dx}\right)^3\;dx ?$$
Thank you for the help =)

REPLY [5 votes]: Here is a solution that does not require the second derivative to exist:
Note, that with $u(x)=y'(x)$, you get the equation $u(x)+(u(x))^3=C$ from which you can conclude that $u(x)$ is a constant, say $D$. But then $y(x)=\int u(x)\,dx=Dx+E$, where $E$ is another constant. 
Now you can insert $y'(x)=D$ into your differential equation to find the constant $D$, $D+D^3=C$ (the general solutions (there are three of them, one is real) do not look too nice). The constant $E$ cannot be determined with the information you have.