Exercises

01

Second-order linear: distinct roots, repeated roots

Find the 2-parameter family of solution curves for the ODEs:

• (a) $2y^{\prime \prime }+6y^{\prime }+2y=0$
• (b) $4y^{\prime \prime }-12y^{\prime }+9y=0$

02

Cancelling imaginary parts

By substituting the formulas for $z_1(t)$ and $z_2(t)$, verify that $y_1(t)=e^{ht}\cos (kt)$ and $y_2(t)=e^{ht}\sin (kt)$.

03

Verifying trigonometric solutions

Verify that $y_1(t)=e^{ht}\cos (kt)$ and $y_2(t)=e^{ht}\sin (kt)$ solve the original ODE.

To do this, first find the correct ODE: expand $a{r}^2+br+c=(r-(h+ki))(r-(h-ki))$ in order to find the appropriate $a,b,c$ for the general case of complex roots.

Then check that $e^{ht}\cos (kt)$ and $e^{ht}\sin (kt)$ are solutions to the ODE $a{y}^{\prime \prime }+b{y}^{\prime }+cy=0$.