In: Math

# General solution of PDE

Find the general solution of the PDE.

$$y^{\prime \prime}-6 y^{\prime}+9 y=0$$

## Solutions

##### Expert Solution

Solution: Given ODE: $$y^{\prime \prime}-6 y^{\prime}+9 y=0$$

This ODE can be written as: $$\left(D^{2}-6 D+9\right) y=0$$

The characteristic equation of the differential equation is:

$$m^{2}-6 m+9=0$$

Factorise the characteristic equation:

$$m^{2}-3 m-3 m+9=0 \Rightarrow(m-3)(m-3)=0$$

Determine the roots of the characteristic equation of the differential equation:

$$m=3,3$$

The roots of the characteristic equation are real and equal,then If the characteristic equation has two real and equal roots, then the general solution is given by:

$$y(x)=\left(C_{1}+C_{2} x\right) e^{m x}$$

then $$y(x)=\left(C_{1}+C_{2} x\right) e^{3 x}$$

The general solution of the Given ODE is:

$$y(x)=\left(C_{1}+C_{2} x\right) e^{3 x}$$

The general solution of the Given ODE is:

$$y(x)=\left(C_{1}+C_{2} x\right) e^{3 x}$$