MATH 20D Lecture 28

Solving Non-Homogeneous Systems

1. Find evals/evects of then form
2. Form fundamental matrix
3. Find the inverse
4. Compute then

Exercise. Find the general solution to the non-homogeneous systems ${\vec{x}}^{\prime }(t)=A\vec{x}(t)+\vec{g}=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\vec{x}+\left[\begin{array}{c}{e}^t\\ -e^t\end{array}\right]$

Example. Find the Laplace transform of $f(t)=\{\begin{array}{l}3t^2-2,t<4\\ e^{5t}+t,4\le t<9\\ 0,\text{ otherwise}\end{array}$

Example. Solve the IVP $y^{\prime \prime }-6y^{\prime }=\delta (t-1)-u_4(t),y(0)=-4,y^{\prime }(0)=0$.

Exercise. Solve the IVP $y^{\prime \prime \prime }-y^{\prime }=0,y(0)=0,y^{\prime }(0)=1,y^{\prime \prime }(0)=-1$.