Q14E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

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

Step-by-Step Solution

Answer

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

Step 1: 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.

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