Problems

01

Solving second-order linear ODEs

Find the complete family of solutions to the following ODEs:

• (a) $y^{\prime \prime }+y^{\prime }-6y=0$
• (b) $y^{\prime \prime }+4y^{\prime }+5y=0$
• (c) $y^{\prime \prime }+y^{\prime }=0$
• (d) $16y^{\prime \prime }-8y^{\prime }+y=0$

02

Solving second-order linear IVPs

Solve the IVP:

• $y^{\prime \prime }-5y^{\prime }+6y=0$, $y(1)=e^2$, $y^{\prime }(1)=3e^2$
• $y^{\prime \prime }+4y^{\prime }+5y=0$, $y(0)=1$, $y^{\prime }(0)=0$

03

Asymptotic behavior

Consider the ODE: $y^{\prime \prime }+b{y}^{\prime }+cy=0$. Let $y(t)$ be a solution.

Show that the following statements are equivalent:

• $b>0$ and $c>0$
• $y(t)\to 0$ as $t\to +\infty$

04

Extending methods to a third-order ODE

Find a complete family of solutions to the third-order ODE $y^{\prime \prime \prime }-3y^{\prime \prime }+4y^{\prime }-2y=0$ using the methods discussed for second-order equations with constant coefficients.

05

Backwards reasoning: find the ODE

Find the ODE that has the following family of solutions: $C_1{e}^{-4t}\cos t+C_2{e}^{-4t}\sin t$.