Answer
Does not have any solution.
Work Step by Step
As we are given that $y''+4y'+29y=0$, $y(0)=1$ and $y(\pi)=-1$
The roots are complex numbers such as $\alpha \pm i \beta$, thus the
general solution is of the form $y=e^{\alpha x}[c_1cos(\beta x)+c_2sin(\beta x)]$
From the given values, $y(0)=1$ and $y(\pi)=-1$, we get $y(0)=1$
$$y(\pi)=-e^{-2\pi}\ne -1$$
This shows that the second boundary condition cannot be satisfied , hence the given problem does not have any solution.
