Question

Solve the following differential equations by the methods discussed above and compare computer solutions.$\left(D^2-4D+13\right)y=0$

Short answer

The solution is$y=e^{2x}\left(c_1{e}^{3ix}+c_2{e}^{-3ix}\right)$

Step-by-step solution

Given information

A differential equation is given as $(D^2-4D+13)y=0$.

Differential equation. 

A differential equation of the form $(D-a)(D-b)y=0,a,b=\alpha \pm \beta i$has general solution

$y=c_1{e}^{\left(\alpha +\beta i\right)x}+c_2{e}^{\left(\alpha -\beta i\right)x}$

Find the solution of the given differential equation.

Suppose that$D=\frac{d}{dx}$

Then

$$\begin{gathered} Dy=\frac{dy}{dx} \\ Dy=y^{\text{'}} \\ {D}^{2}y=\frac{d}{dx}\left(\frac{dy}{dx}\right) \\ \frac{d^{2}y}{d{x}^2}=y^{\text{'}\text{'}} \end{gathered}$$

Given differential equation is $\left(D^2-4D+13\right)y=0$.

Auxiliary equation is

$$\begin{gathered} D^2-4D+13=0 \\ D=\frac{-(-4)\pm \sqrt{{(-4)}^2-4·1·13}}{2·1} \\ D=\frac{4\pm 6i}{2} \\ D=2+3i \\ D=2-3i \end{gathered}$$

These are unequal roots.

$\Rightarrow D=2+3i$and$D=2-3i$

Differential equation becomes -

$\Rightarrow [D-(2+3i\left)\right][D+(2-3i\left)\right]y=0$

These are separable equation with solution

$y=c_1{e}^{(2+3i)x}\text{and}y=c_2{e}^{(2-3i)x}$

The general solution will be given by

$$\begin{gathered} y=c_1{e}^{(2+3i)x}+c_2{e}^{(2-3i)x} \\ y=e^{2x}\left(c_1{e}^{3ix}+c_2{e}^{-3ix}\right) \end{gathered}$$