I have the following DE: $$ f'' + p(t)f' + q(t) f = 0 $$
where $$ p(t)= \frac{a}{t} \text{ and } q(t) = b\left(-\frac{1}{t^2} + \frac{\epsilon}{t^3} \right) $$ with $a,b, \epsilon \in \mathbb{R}$ and $0<\epsilon \ll 1$. Now, if $\epsilon = 0$, the above equation is just a Cauchy-Euler equation that I have solved exactly. However, for $\epsilon \neq 0$ we see that $t=0$ is an irregular singular point and as such I am interested in an asymptotic approximation of the solution in the region $t \in (0, 1)$.
I am not very familiar with asymptotic analysis, but I think I might be able to use the method of dominant balance to write down the asymptotic behavior as $t \rightarrow 0^+$. However, I am curious if there is some better way to do this given that I have the solution for the $\epsilon = 0$ case. Is there a way to get an approximate solution in powers of $\epsilon$, for example?
Any pointers or resources would be greatly appreciated - just something to get me headed down the right path.
EDIT: Robert Israel has pointed out below that an exact solution exists for this differential equation. Nonetheless, I am still interested in the approximate solution. The more general question involves equations with $$ q(t) = b \left( -\frac{1}{t^2} + \frac{\epsilon_1}{t^3} + \frac{\epsilon_2}{t^4} \right) $$ Or even higher orders of $t^{-n}$, and where $\epsilon_i$ are small parameters.
Your differential equation has closed-form solutions
$$ \eqalign{f \left( t \right) &=c_{{1}}{t}^{1/2-a/2}{{ J}_{-\sqrt {{a}^{2}-2\,a +4\,b+1}}\left(2\,{\frac {\sqrt {b\epsilon}}{\sqrt {t}}}\right)}\cr &+c_{{2 }}{t}^{1/2-a/2}{{ Y}_{-\sqrt {{a}^{2}-2\,a+4\,b+1}}\left(2\,{\frac {\sqrt {b\epsilon}}{\sqrt {t}}}\right)}} $$
where $J$ and $Y$ are Bessel functions. The asymptotics of these functions as $t \to 0$ are known.
EDIT: The transformation $f(t) = t^{1/2 - a/2} g(2 \sqrt{b \epsilon/t})$ makes the equation into $$ s^2 g''(s) + s g'(s) + (s^2 + 2a - a^2 - 4 b - 1) g(s) = 0$$
Your second differential equation has closed-form solutions as well:
$$\eqalign{f \left( t \right) &=c_{{1}}{t}^{-a/2+1}{{ M}_{{\frac {-i\sqrt {b} \epsilon_{{1}}}{2\sqrt {\epsilon_{{2}}}}},\,\sqrt {{a}^{2}-2\,a+4 \,b+1}/2}\left({\frac {2\,i\sqrt {\epsilon_{{2}}}\sqrt {b}}{t}}\right)}\cr &+ c_{{2}}{t}^{-a/2+1}{{ W}_{{\frac {-i\sqrt {b}\epsilon_{{1}}}{2 \sqrt {\epsilon_{{2}}}}},\,\sqrt {{a}^{2}-2\,a+4\,b+1}/2}\left({ \frac {2\,i\sqrt {\epsilon_{{2}}}\sqrt {b}}{t}}\right)}} $$ where $M$ and $W$ are Whittaker functions.