Chapter 7.9, Problem 13P

a.

To determine

An equivalent non-homogeneous system of first order linear differential equations for $x_1=y,x_2=y^{\prime }$ for the differential equation $y^{″}-5y^{\prime }+6y=2e^t,y_1=e^{2t},y_2=e^{3t}$.

b.

To determine

To show: That $x^{\left(1\right)}={\left(\begin{array}{cc}{y}_1& y_1{}^{\prime }\end{array}\right)}^T\text{ and }{x}^{\left(2\right)}={\left(\begin{array}{cc}{y}_2& y_2{}^{\prime }\end{array}\right)}^T$ are the solution of the homogeneous system of differential equations corresponding to the system found in the sub part (a).

c.

To determine

The variation of parameters that have to be satisfied for $y=y_1\left(t\right)u_1\left(t\right)+y_2\left(t\right)u_2\left(t\right)$, to be a particular solution of the given non-homogeneous differential equation.

d.

To determine

The variation of parameters that have to be satisfied for $x=\psi \left(t\right)u\left(t\right)$, to be a particular solution of the given non-homogeneous differential equation.

e.

To determine

To show: That the equations obtained in c and d are equivalent using the definitions mentioned in a and b.