91Ó°ÊÓ

What is meant by first order, linear, nonhomogeneous vector differential equation?

Give an example of a homogeneous and a nonhomogeneous system of first order linear differential equations in scalar and matrix form.

Solve $$ \begin{aligned} &\qquad \begin{aligned} \mathrm{x}+\mathrm{Dy}=\cos \mathrm{t}+\sin \mathrm{t} \\ \mathrm{Dx}+\mathrm{y}=\cos \mathrm{t}-\sin \mathrm{t} \end{aligned} \\ &\text { where } \mathrm{D}=(\mathrm{d} / \mathrm{dt}) \end{aligned} $$

Solve $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=\mathrm{y} \\ &(\mathrm{dy} / \mathrm{dt})=-2 \mathrm{x}+3 \mathrm{y} \end{aligned} $$

Solve \((d x / d t)=3 x+2 y\) \((d y / d t)=-5 x+y\)

Solve $$ \begin{aligned} &(d x / d t)=4 x-y \\ &(d y / d t)=x+2 y \end{aligned} $$

Solve $$ \begin{aligned} &(d x / d t)=6 x-3 y \\ &(d y / d t)=2 x+y \end{aligned} $$

Solve $$ \begin{aligned} &2 y_{1}^{\prime}-3 y_{2}^{\prime}=2 e^{2 x} \\ &y_{1}^{\prime}-2 y_{2}^{\prime}=0 \end{aligned} $$ subject to the initial conditions $$ \mathrm{y}_{1}(0)=2 $$ $$ \text { and } \mathrm{y}_{2}(0)=1 $$

Determine the general solution of the nonhomogeneous system $$ \begin{aligned} &y^{\prime \prime}(x)-y(x)+5 v^{\prime}(x)=x \\ &2 y^{\prime}(x)-v^{\prime \prime}(x)+4 v(x)=2 \end{aligned} $$ with independent variable \(\mathrm{x}\) and dependent variables \(\mathrm{y}(\mathrm{x})\) and \(\mathrm{v}(\mathrm{x})\).

Find the general solution of the system of simultaneous linear equations $$ \begin{aligned} &\mathrm{y}^{\prime}=\mathrm{y}-3 z \\ &\mathrm{z}^{\prime}=2 \mathrm{y}-4 \mathrm{z} \end{aligned} $$