Problems

01

Complex eigenvalues

Find the complete family of solutions for the system ${\mathbf{y}}^{\prime }=\left(\begin{array}{rrr}1& 2& -1\\ 0& 1& 1\\ 0& -1& 1\end{array}\right)\mathbf{y}$.

02

Generalized eigenvector

Find the complete family of solutions for the system ${\mathbf{y}}^{\prime }=\left(\begin{array}{rr}1& -1\\ 1& 3\end{array}\right)\mathbf{y}$.

03

Cauchy-Euler system

The Cauchy-Euler equation is given by $a{t}^2{y}^{\prime \prime }+bt{y}^{\prime }+cy=0$.

• (a) Show how to convert this $2^{\text{nd}}$ order linear equation to a system of the form $t{\mathbf{y}}^{\prime }=A\mathbf{y}$. (Hint: $y_0=y/t,y_1=y^{\prime }$.)
• (b) Show that the solutions to this system have the form $\mathbf{y}(t)=t^{\lambda }\mathbf{v}$, for $\mathbf{v}$ an eigenvector of $A$ with eigenvalue $\lambda$.
• (c) Solve the system $t{\mathbf{y}}^{\prime }=\left(\begin{array}{rr}-1& -1\\ 9& -1\end{array}\right)\mathbf{y}$.

04

Repeated roots

• (a) Consider the generic equation with repeated roots: $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$. (Repeated roots implies that $b^2=4ac$ and $r=-\frac{b}{2a}$.) Solve this ODE by converting it to a $2$-variable system and using the matrix exponential technique.
• (b) Solve the $3^{\text{rd}}$ order ODE given by $y^{\prime \prime \prime }-6y^{\prime \prime }+12y^{\prime }-8y=0$ by converting it to a $3$-variable system and using the matrix exponential technique. (Hint: it works well to use ${\mathbf{e}}_1$, $A_2{\mathbf{e}}_1$, and $A_2^2{\mathbf{e}}_1$ for your generalized eigenvectors.) (Note: you haven’t been able to solve with a triple repeated root before!)
• (c) Verify that $e^{2t}$, $t{e}^{2t}$, and $t^2{e}^{2t}$ each solve the equation in (b). Translate these functions into the vector-valued format for the system you created in (b). Express these vector solutions in terms of the complete family you found with the matrix exponential technique in (b) by choosing appropriate parameter constants.