Question

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

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

Short 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-by-step solution

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.

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