Differential Equations

1.

Check if the following functions are solutions to the given EQ.

1. Check directly if $y_{1}=2e^{5t}$ is a solution or not to $y^{\prime\prime}-6y^{\prime}+5y=0\text{.}$
2. Check directly if $y_{2}=2e^{t}$ is a solution or not to $y^{\prime\prime}-6y^{\prime}+5y=t\text{.}$

2.

Recall that if $y(t)=e^{rt}$ is a solution to the ODE given by

for constant $a,b,c$ where $a\ne0\text{,}$ then the exponent $r$ in front the $t$ must be a solution to the characteristic equation $ar^{2}+br+c=0\text{.}$

By yourself, rederive that if $y(t)=Ae^{rt}$ is a solution to the equation above, then the number $r$ must satisfy the characteristic equation $ar^{2}+br+c=0$ or $A=0\text{.}$

3.

Use the method given in Section 3.2 to find the general solution to

4.

Use the method given in Section 3.2 to find the general solution to

5.

Use the method given in Section 3.2 to find the particular solution to the IVP