hint for solving in $t$, the following equation $t^2e^{a^2\,/t} + a^2e^t-12at=0$

Question

Please give me a hint in order to solve in t, the following equation:

$$t^2e^{a^2\!/t} + a^2e^t-12at=0$$

where $a=\ln(4)$.

Answer 1

Hint: plug in $t:=a\cdot x$ for a new unknown $x$. This simplifies the equation a bit and (provided I didn't make a computation error) if you plug in $a=\ln(4)$ and then try some small integers for $x$ you will find a solution.

Edit: You already found $x^2*e^{a/x}+e^{ax}-12*x=0$. Now plug in $a=\ln(4)$ gives $$x^24^{1/x} + 4^x = 12x$$ Pluggin in $x=2$ gives a solution.

Note that this uses being lucky with your specific equation. If you replace the $12$ by a $13$ I don't think there is any way to solve this algebraically.