Question

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

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

Short 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-by-step solution

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

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