Solve the following initial value problem: $$ \begin{aligned} \mathrm{y}^{\prime \prime}+2 \mathrm{y}^{\prime}+\mathrm{y} &=0 \\ \mathrm{y}(1) &=0 \\ \mathrm{y}^{\prime}(1) &=2 . \end{aligned} $$

Solve the following initial value problem: $$ \begin{aligned} &x^{\prime}=4 x+y \\ &y^{\prime}=3 x+2 y \end{aligned} $$ Initial conditions: \(\quad \mathrm{x}(0)=-1\) $$ \mathrm{y}(0)=7 $$

Find the general solution of the following system of differential equations: $$ \begin{aligned} &3\\{(d x / d t)+(d y / d t)\\}=\sin t \\ &(d x / d t)+3(d y / d t)=\cos t \end{aligned} $$

Let $$ \begin{aligned} &(d x / d t)=a x+b y \\ &(d y / d t)=c x+d y . \end{aligned} $$ Discuss the nature of the critical point of \((1)\) when the roots of the associated characteristic equation are: a) complex; b) pure imaginarv

Solve the following initial value problem: $$ \begin{gathered} \mathrm{y}_{1}^{\prime}=\mathrm{y}_{1}+\mathrm{y}_{2} \\ \mathrm{y}_{2}^{\prime}=4 \mathrm{y}_{1}-2 \mathrm{y}_{2} \\ \text { Initial conditions: } \mathrm{y}_{1}(0)=1, \mathrm{y}_{2}(0)=6 \end{gathered} $$

The general solution of the system $$ \begin{aligned} &\mathrm{d} \mathrm{x} / \mathrm{dt}=\mathrm{ax}+\mathrm{by} \\ &\mathrm{dy} / \mathrm{dt}=\mathrm{cx}+\mathrm{dy} \end{aligned} $$ is $$ \mathrm{x}=\mathrm{A}_{1} \mathrm{e}^{(\mathrm{r}) 1(\mathrm{t})}+\mathrm{A}_{2} \mathrm{e}^{(\mathrm{r}) 2(\mathrm{t})}, \mathrm{y}=\mathrm{B}_{1} \mathrm{e}^{(\mathrm{r}) 1(\mathrm{t})}+\mathrm{B}_{2} \mathrm{e}^{(\mathrm{r}) 2(\mathrm{t})} $$ Discuss, with the aid of examples, the nature of the critical point when: a) \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) are real, and of the same sign, but \(\mathrm{r}_{1} \neq \mathrm{r}_{2}\); b) \(\mathrm{r}_{1}\) and \(\mathrm{r}_{2}\) are real, and of opposite signs, and \(\mathrm{r}_{1} \neq \mathrm{r}_{2}\).