Find the value(s) of the constants C and λ

Question

Find the value(s) of the constants C and λ for which

$\theta_1$(t)=$e^{t\lambda}$ and $\theta_2$(t)=$Ce^{t\lambda}$

is a solution to the equations of motion

$$2\ddot\theta_1+\ddot\theta_2 + \frac{2g}{1}\theta_1=0$$

$$\ddot\theta_1+\ddot\theta_2 + \frac{g}{1}\theta_2=0$$

I began by making them equal to each other to see if that would help but all I got was

$$\ddot\theta_1+\frac{2g}{1}\theta_1= \frac{g}{1}\theta_2$$

and I don't know where to go from here or if it is even in the right direction.

please feel free to add tags I wasn't certain what one would fit this topic all help is really appreciated.

Answer 1

Why not just insert them into the DEs: $(e^{\lambda t})''=\lambda^2 e^{\lambda t}$ and $(Ce^{\lambda t})''=C\lambda^2e^{\lambda t}$, so we have: $$2\lambda^2e^{\lambda t}+C\lambda^2e^{\lambda t}+2ge^{\lambda t}=0\implies(2+C)\lambda^2+2g=0\tag1$$ $$\lambda^2e^{\lambda t}+C\lambda^2e^{\lambda t}+gCe^{\lambda t}=0\implies(1+C)\lambda^2+gC=0\tag2$$

Solve this pair and you have your $\lambda$ and $C$.