Question

Find the general solutions of the following equations and compare computer solutions.

${\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{3}{\mathit{y}}^{\mathbf{\text{'}}\mathbf{\text{'}}}\mathbf{-}\mathbf{9}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{-}\mathbf{5}\mathit{y}\mathbf{=}\mathbf{0}$

Short answer

The general solution is,

$y=c_1{e}^{5x}+(c_{3}x+c_5)e^{-x}$

Step-by-step solution

Given information from question 

Given equation is$y^{\text{'}\text{'}\text{'}}-3y^{\text{'}\text{'}}-9y^{\text{'}}-5y=0$

Differential equation

Differential equation:

A differential equation is a formula that connects the derivatives of one or more unknown functions. Functions are used to represent physical quantities, derivatives are used to characterise their rates of change, and differential equations are used to define a relationship between them in applications.

 Step 3: Calculate the general solution

The auxiliary equation is,

$\begin{array}{l}(D^3-3D^2-9D+5)y=0\\ (D-5)(D+1)(D+1)y=0\end{array}$

As it can see from the auxiliary equation it has only two roots, therefore something similar to the way of finding the solution. Now start by finding the solution for the equation

$\begin{array}{c}\frac{dy}{dx}=5y\\ y=c_1{e}^{5x}\end{array}$

Use the second root to find the second solution  

Then using the second root, which is,$D=-1$then find the second solution of this differential equation

$\begin{array}{c}\frac{dy}{dx}=-y\\ y=c_2{e}^{-x}\end{array}$

Lastly, let$u=(D+1)y$to find the third solution, therefore

$\begin{array}{c}(D+1)u=0\\ \frac{du}{dx}=-u\\ u=c_3{e}^{-x}\end{array}$

Substitutes back, to get

$\begin{array}{l}(D+1)y=c_3{e}^{-x}\\ \frac{dy}{dx}+y=c_3{e}^{-x}\end{array}$

This is first order differential equation, and to solve:

$\begin{array}{c}I-\int dx=x\\ e^l=e^x\\ y{e}^l=\int \left(c_3{e}^{-x}\right)e^{x}dx\\ =c_{3}x+c_4\end{array}$

Then, the general solution of the differential equation is.$y=c_1{e}^{5x}+(c_{3}x+c_5)e^{-x}$