Question

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

$y^{\left(4\right)}+2y\text{'}\text{'}\text{'}+10y\text{'}\text{'}+18y\text{'}+9y=0$

Short answer

The general solution for the differential equation with x as the independent variable is $y\left(x\right)=c_1{e}^{-x}+c_{2}x{e}^{-x}+c_3\cos 3x+c_4\sin 3x$

Step-by-step solution

Auxiliary equation:

Consider the equation .$y^{\left(4\right)}+2y\text{'}\text{'}\text{'}+10y\text{'}\text{'}+18y\text{'}+9y=0$

The associated auxiliary equation is $r^4+2r^3+10r^2+18r+9={(r+1)}^2(r^2+9)=0$ which has$r=-1$ as a double solution and$r=-3i$ and$r=3i$ as a simple solutions.

General solution:

The general solution to the given equation is given by

$y\left(x\right)=c_1{e}^{-x}+c_{2}x{e}^{-x}+c_3\cos 3x+c_4\sin 3x$

Hence the final solution is $y\left(x\right)=c_1{e}^{-x}+c_{2}x{e}^{-x}+c_3\cos 3x+c_4\sin 3x$