I understand how to solve 2 element transient but am having some problems with 3+ element transients.
Specifically I'm trying to solve this equation:
$$737280 e^{-2400t}-576000 e^{-1500t} + 46080 e^{-600t}=0$$
Whenever I use WolframAlpha it always gives me a complex answer even though I know I real answer exists.
I'm also curious if a general solution to this form exists. E.g.
$$Ae^{at}+Be^{bt}+Ce^{ct}=0$$
If $e^{300t}=x$, then your equation can be written
$$Ax^8+Bc^5+Cx^2=x^2(Ax^6+Bc^3+C)=0$$
Since $x>0$ for all $t$, you have just to solve
$$Ax^6+Bc^3+C=0$$
But with $y=x^3=e^{900t}$, it's just
$$Ay^2+By+C=0$$
Now you should be able to solve. Remember that $y$, which is an exponential, is always positive.
With your coefficients, the equation can even be simplified a bit more, since
$$737280y^2-576000y+46080=23040\cdot(32y^2-25y+2)$$