Implicit Differentiation of $xe^{tx} = C$.

Question

Show that any function $x=x(t)$ that satisfies the equation $xe^{tx} = C$ is a solution of the differential equation $(1+tx)x'=-x^2$. In the answer they state that you should Differentiate $xe^{tx} = C$ implicitly w.r.t. $t$. But I do not know how to do such an implicit Differentiation. Could someone help me out a bit?

Answer 1

Consider:

$$x(t)e^{tx(t)}=C$$

If we differentiate this with respect to $t$ then we get:

$$x'(t)e^{tx(t)}+(tx'(t)+x(t))x(t)e^{tx(t)}=0$$

Which implies that:

$$-x^2(t)=x'(t)(1+tx(t)).$$