Answer
$${y}=\frac{5+\sqrt13}{2\sqrt13}\exp^{-\frac{5-\sqrt13}{2}x}+\frac{\sqrt{13}-5}{2\sqrt13}\exp^{-\frac{5+\sqrt13}{2}x}$$
Work Step by Step
$$y''+5y'+3y=0,\quad{y}(0)=1,\quad{y'}(0)=0$$Let $y=e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$.
$${\lambda}^2+5{\lambda}+3=0$$
$$\lambda_{1,2}=\frac{-5\pm\sqrt{25-12}}{2}=\frac{-5\pm\sqrt13}{2}$$
The general solution is ${y}=c_{1}\exp^{-(\frac{5-\sqrt13}{2})x}+c_{2}\exp^{-(\frac{5+\sqrt13}{2})x}$. Substituting in the constraints, we obtain
$c_{1}+c_{2}=1$ and $-(\frac{5-\sqrt13}{2})c_{1}-(\frac{5+\sqrt13}{2})c_{2}=0\Rightarrow{c}_{1}=\frac{5+\sqrt13}{\sqrt{13}-5}c_{2}$.
$$\therefore\frac{2\sqrt13}{\sqrt{13}-5}c_{2}=1\Rightarrow{c}_{2}=\frac{\sqrt{13}-5}{2\sqrt13}\Rightarrow{c}_{1}=\frac{5+\sqrt13}{2\sqrt13}$$
$$\therefore{y}=\frac{5+\sqrt13}{2\sqrt13}\exp^{-\frac{5-\sqrt13}{2}x}+\frac{\sqrt{13}-5}{2\sqrt13}\exp^{-\frac{5+\sqrt13}{2}x}$$
