A friend of mine has given me a nice problem to crack recently. I have not been able to solve it despite multiple attempts:
For a real function $x(t)$ and real constant $a$, it is known that $$ax(t)-\frac{d}{dt}x(t)=-\exp(at).$$ Given that $x(0)=0$ and $x(t_0)=1/a$, find the real constant $t_0$.
I suspect that if I can solve the differential equation, then solving for $t_0$ should be easy. Now, Wolfram Alpha gives me the general solution quite immediately, but of course it does not give the steps. I was wondering what the analytical way to solve it by hand was. Thanks in advance for the input!
From
$$x'-ax=e^{at}$$
we draw $$(x'-ax)e^{-at}=(xe^{-at})'=1$$
and by integration, using the initial condition,
$$x\,e^{-at}=t.$$
Now
$$\frac1ae^{-at_0}=t_0$$ is equivalent to
$$at_0e^{at_0}=1$$
or
$$at_0=W(1).$$