Question

Solve the following differential equations by the methods discussed above and compare computer solutions.

$\left(D^2-5D+6\right)y=0$

Short answer

The solution is $y=c_1{e}^{2x}+c_2{e}^{3x}$.

Step-by-step solution

Given information

A differential equation is given as $\left(D^2-5D+6\right)y=0$.

Differential equation. 

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

$y=c_1{e}^{ax}+c_2{e}^{bx}.$

Find the solution of 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 $\left(D^2-5D+6\right)y=0$.

Auxiliary equation is-

$$\begin{gathered} D^2-5D+6=0 \\ (D-2)(D-3)=0 \\ D=2 \\ D=3 \end{gathered}$$

These are unequal roots.

$\Rightarrow D=2,D=3$

Differential equation becomes

$\Rightarrow (D-2)(D-3)y=0$

These are separable equation with solution

$y=c_1{e}^{2x}\text{and}y=c_2{e}^{3x}$

The general solution will be given by $y=c_1{e}^{2x}+c_2{e}^{3x}$.