Question

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

$\mathbf{(}{\mathbf{D}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathbf{D}\mathbf{+}\mathbf{6}\mathbf{)}\mathit{y}\mathbf{=}\mathbf{0}$

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.$(D^2-5D+6)y=0$

Differential equation. 

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

$\mathit{y}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{\mathbf{a}\mathbf{x}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\mathbf{b}\mathbf{x}}\mathbf{.}$

Find the solution of given differential equation. 

Supposethat.$D=\frac{d}{dx}$

Then

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

Given differential equation is.$(D^2-5D+6)y=0$

Auxiliary equation is-

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

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 givenby.$y=c_1{e}^{2x}+c_2{e}^{3x}$