Q10E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Find a general solution for the differential equation with x as the independent variable:

$y\text{'}\text{'}\text{'}+3y\text{'}\text{'}-4y\text{'}-6y=0$

Step-by-Step Solution

Answer

The general solution for the differential equation with x as the independent variable is  $y\left(x\right)=c_1{e}^{-x}+c_2{e}^{(-1-\sqrt{7})x}+c_3{e}^{(-1+\sqrt{7})x}$

Step 1: Auxiliary equation:

Consider the equation . $y\text{'}\text{'}\text{'}+3y\text{'}\text{'}-4y\text{'}-6y=0$

The associated auxiliary equation is  which $r^3+3r^2-4r-6=(r+1)(r^2+2r-6)=0$ has  $r=-1,r=-1-\sqrt{7}$  and   $r=-1+\sqrt{7}$ as solutions

Step 2: General solution:

The general solution to the given equation is given by 

 $y\left(x\right)=c_1{e}^{-x}+c_2{e}^{(-1-\sqrt{7})x}+c_3{e}^{(-1+\sqrt{7})x}$

Hence the final solution is  $y\left(x\right)=c_1{e}^{-x}+c_2{e}^{(-1-\sqrt{7})x}+c_3{e}^{(-1+\sqrt{7})x}$