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:

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

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