Q9E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

$u\text{'}\text{'}\text{'}-9u\text{'}\text{'}+27u\text{'}-27u=0$

Step-by-Step Solution

Answer

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

Step 1: Auxiliary equation:

In the equation, $r^3-9r^3+27r-27=0$  , we recognize a complete cube, namely, ${(r-3)}^3=0$ . Thus, it has just one root x = 3 of multiplicity three.

Step 2: General solution:

The general solution to the given differential equation is given by  

$u\left(x\right)=c_1{e}^{3x}+c_{2}x{e}^{3x}+c_3{x}^2{e}^{3x}$

Hence the final solution is  $u\left(x\right)=c_1{e}^{3x}+c_{2}x{e}^{3x}+c_3{x}^2{e}^{3x}$