91Ó°ÊÓ

A differential equation is given as.$D(D+5)y=0$

-Auxiliary equation:

Auxiliary equation is an algebraic equation of degreeupon which depends the solution of a given nth-order differential equation or difference equation.

-General form of the auxiliary equation$\mathbf{(}\mathit{D}\mathbf{−}\mathit{a}\mathbf{)}\mathbf{(}\mathit{D}\mathbf{−}\mathit{b}\mathbf{)}\mathbf{=}\mathit{k}{\mathit{e}}^{\mathbf{c}\mathbf{x}}$

First, write the auxiliary equation,

$\begin{array}{c}(D^2−16)y=40e^{4x}\\ (D+4)(D−4)y=40e^{4x}\end{array}$

The complementary solution is corresponding to the same differential equation but with zero wight hand side, that is

$(D+4)(D−4)y=0$

The solution for this differential equation is in the form of eq.(5.11) because the roots of the auxiliary equation are not equal. That is,$y_c=A{e}^{−4x}+B{e}^{4x}$

Next, the particular- solution could be founded by successive integration of two first order equations (need to omit integration constant each time to get the particular- solution). Let

$u=(D−4)y$

therefore, the differential equation becomes

$\begin{array}{c}(D+4)u=40e^{4x}\\ u^{\text{'}}+4u=40e^{4x}\end{array}$

This is first order differential equation, and solve it by making use of eq.(3.4) and eq.(3.9) (remember, need to drop integration constants), that is

$\begin{array}{c}I=∫4dx\\ I=4x\text{ }\\ e^I=e^{4x}u{e}^I\\ =∫\left(40e^{4x}\right)e^{4x}dx\end{array}$

Solve further the equation

$\begin{array}{l}=5e^{8x}u\\ =5e^{4x}\end{array}$

Now, substitute this result in$u=(D−4)y$to get

$5e^{4x}=y^{\text{'}}−4y$

which has become (again) a first order differential equation. find the solution of such and equation as follow

$\begin{array}{c}I=−∫4dx\\ =−4x{y}_p{e}^I\\ =∫\left(5e^{4x}\right)e^{−4x}dx\\ =5x{y}_p\end{array}$

Further solve the equation

$=5x{e}^{4x}$

Therefore, the general solution of the differential equation is,$y=y_c+y_p$ that is

$y=A{e}^{−4x}+B{e}^{4x}+5x{e}^{4x}$

,,$\left[\text{DSolve}\right[\text{y},\text{x}\left[\text{x}\right]−16\text{y}\left[\text{x}\right]==40\text{E}\left(4\text{x}\right),\text{y}\left[\text{x}\right]$ get the same answer as above