Solving the differential equation $\frac{dx}{dt} = \frac{a^2}{(x+t)^2}$ by substitution I'm struggling with the following differential equation:
$$\frac{dx}{dt} = \frac{a^2}{(x+t)^2}.$$ 
Firstly I used a substitution: $u=x+t$ but as a result I get 
$$t+c = u - a \cdot \arctan\left(\frac{u}{a}\right),$$ 
from which I can't get $u$. I tried some other substitutions, but none of them gave me a result. Can somebody help me with finding the right substitution?

Let
$$u=x+t$$
By Chain Rule
$$\frac{du}{dx}\frac{dx}{dt}=\frac{du}{dt}=\frac{dx}{dt}+1$$
Thus, 
$$\frac{du}{dx}\frac{a^2}{u^2}=\frac{a^2}{u^2}+1$$
$$a^2\frac{du}{dx}=u^2+a^2$$
$$\frac{a^2}{a^2+u^2}du=dx$$
Integrate both sides
$$a\arctan \frac{u}{a}=x+C$$
$$x+t=u=a\tan(\frac{x+C}{a})$$
$$t=a\tan(\frac{x+C}{a})-x$$
