In: Math
Find the general solution of the PDE.
\(y^{\prime \prime}-6 y^{\prime}+9 y=0\)
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:
\(y(x)=\left(C_{1}+C_{2} x\right) e^{3 x}\)