Differential Equations: Problem Set 8

Show that if $x_1\left(t\right)$ and $x_2\left(t\right)$ are solutions of $x^{\prime \prime }+p\left(t\right)x^{\prime }+q\left(t\right)x=0$, then so is $x\left(t\right)=c_1{x}_1+c_2{x}_2$.

Find two constants ${\lambda }_1$ and ${\lambda }_2$ such that $e^{{\lambda }_{1}t}$ and $e^{{\lambda }_{2}t}$ are each solutions of $x^{\prime \prime }+3x^{\prime }+2x=0$. Then verify that $c_1{e}^{{\lambda }_{1}t}+c_2{e}^{{\lambda }_{2}t}$ is also a solution.

Show that if $x_1\left(t\right)$ and $x_2\left(t\right)$ are solutions of $x^{\prime \prime }+p\left(t\right)x^{\prime }+q\left(t\right)x=0$, and there exists $t_0$ such that $W(x_1,x_2)\left(t_0\right)\ne 0$, then $W(x_1,x_2)\left(t\right)\ne 0$ for all $t$.

(3.1) 10, 12, 14, 16, 18