Simple Ordinary Differential Equation As simple as this should be, I can not seem to solve it. I can't classify its type and thus figure out how to solve it. It's the only ODE in my problem sheet that I can't solve (embarrassing). 
$$t\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)=x+\sqrt{t^2+x^2}$$
 A: Hint, let:
$$x = t v(t) \rightarrow x' = v + t v'$$
Substituting into the original equation, doing some algebra and rearranging, eventually leads to the integration:
$$\int \dfrac{dv}{\sqrt{v^2+1}} = \int \dfrac{1}{t} dt$$
Can you take it from there?
A: This differential equation is an example of a homogeneous differential equation. The reason that change of variables works is because that is the standard technique for solving Euler-homogeneous differential equations. You should always check if a differential equation is exact (poincare's lemma), if non-exact then find an integrating factor (Frobenius theorem) & theoretically the problem can always be solved. However there are shortcuts or more direct methods in special cases such as separable equations, euler-homogeneous equations, linear equations, linear fractional equations, then a bunch of equations with names attached. After that unless you can find a change of variables (Lie theory) you have to use some form of approximation technique, at least in the case of first degree ode's of first order. You shouldn't really think of it as change of variables technique because it only really applies in the homogeneous case, & homogeneity is used for studying higher degree ode's as well hence it has some importance on it's own. Here's a good book.
