How do I find Wronskian for $y(t) = (c_1, tc_2, t^2c_3)e^{at}$.

Question

If I try to use the formula it becomes very difficult to compute: $$y'(t) = ac_1e^{at} + atc_2e^{at}+ c_2e^{at} + at^2c_3e^{at} + 2tc_3e^{at}$$ and then continue with $y''$ and use the formula it becomes very complicated. Is there an easier way of doing this?

Answer 1

From the solution: $$y(t) = (c_1, tc_2, t^2c_3)e^{at}$$ you can deduce the characteristic polynomial: $$(r-a)^3=0$$ $$r^3-3ar^2+3a^2r-a^3=0$$ Then you can deduce the differential equation: $$y'''-3ay''+3a^2y'-a^3y=0$$ $$y'''+p_1y''+p_2y'+p_3y=0$$ Then you can deduce the Wronskian using Abel's Identity. $$W=c \exp \left (-\int p_1(t)dt \right)$$ Here $p_1=-3a$. $$W(t)=ce^{3at}$$