I am given the equation:
$$y''' - 7y'' + 16y' + 10y = 0$$
where a root of the characteristic equation is $\lambda= -1$.
I gathered from this that $y=e^{-x}$ is a solution of the third order ODE.
I am asked to find the general solution of this ODE, so, I am trying to find another solution/s to the ODE, however, I am asked to find this solution without technology.
How can I solve the above cubic without a calculator? From the wording of the question, it sounds like I should use $y = e^{-x}$, but I am not sure how that can assist me.
I plugged the corresponding values in the ODE my calculator out of curiosity, and found that $\lambda = -3 + I$ and $\lambda = -3 - I$ are solutions to the characteristic equation..
Thank you for your time.
I will assume the your equation is actually
$$ y''' \color{red}{+}7y'' + 16y' + 10y = 0 $$
Since $\lambda=-1$ does not solve the one you wrote.
Have you tried solving the characteristic polynomial first? Knowing one of the roots, it can be factored to
$$ (\lambda + 1)(\lambda^2 + 6\lambda + 10) = 0 $$