Examples

Repeated roots

Repeated roots

Problem: Solve the ODE: $y^{\prime \prime }-6y^{\prime }+9y=0$.

Solution: We first solve $r^2-6r+9=(r-3{)}^2=0$. So we have $r_1=r_2=3$. Then $y_1(t)=e^{3t}$ is one solution, and $y_2(t)=t{e}^{3t}$ is another solution. The complete family of solutions:

$$y(t)=C_1{e}^{3t}+C_{2}t{e}^{3t}.$$

Check this complete family by plugging it into the ODE!

Second-order linear, complex roots, pure real solutions

Second-order linear, complex roots, pure real solutions

Problem: Solve the ODE: $y^{\prime \prime }-2y^{\prime }+5y=0$.

Solution: We first solve $r^2-2r+5=0$ and find $r_1=1+2i$ and $r_2=1-2i$. We therefore have complex solutions

$$z_1(t)=e^t(\cos (2t)+i\sin (2t)),z_2(t)=e^t(\cos (2t)-i\sin (2t))$$

and real solutions

$$y_1(t)=e^t\cos (2t),y_2(t)=e^t\sin (2t).$$

The complete family of solutions is therefore:

$$y(t)=C_1{e}^t\cos (2t)+C_2{e}^t\sin (2t).$$