Show that the system of differential equation has no non-trivial solution Consider the following system of differential equation with constant coefficients
$$\frac{dx}{dt}=a_1x+b_1y$$
$$\frac{dy}{dt}=a_2x+b_2y$$
Assume the roots of the characteristic equations are real and equal.
Assume that the system is not such $a_1=b_2\neq 0 $ and $ a_2=b_1=0 $.
Show that there exists no non-trivial solution of the form
$$x=Ate^{\lambda t}$$
$$y=Bte^{\lambda t}$$
which is linearly independent of the solution
$$x=Ae^{\lambda t}$$
$$y=Be^{\lambda t}$$
My work:
I try to differentiate and substitute into the differential equation,
Find out the determinant which is very messy.
I am not sure how to show this? Any simpler way?

Since $$(Ate^{\lambda t})'=Ae^{\lambda t}+At\lambda e^{\lambda t}$$ must be equal to $$a_1x+b_1y=(a_1A+b_1B)te^{\lambda t},$$ we find that $A=0$ (note that $e^{\lambda t}$ and $te^{\lambda t}$ are linearly independent).
Similarly, you get $B=0$ substituting in the other equation. So there exist no nontrivial solution of that form.
