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tonny07

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Consider the following differential ordinary equation
y'' - 5y' + 6y = -3sin (2x).
a) Write the suitable characteristic equation.
b) Determine the root of the characteristic equation.
c) Write the solution of the homogenous equation.
d) Determine the general solution of the differential equation.
 
There are various ways to tackle this. Have you heard of Variation of Parameters?.

The characteristic equation can be found by factoring the quadratic:

\(\displaystyle m^{2}-5m+6=0\) and finding its roots.
 
Consider the following differential ordinary equation
y'' - 5y' + 6y = -3sin (2x).
a) Write the suitable characteristic equation.
b) Determine the root of the characteristic equation.
c) Write the solution of the homogenous equation.
d) Determine the general solution of the differential equation.
Click to expand...

Did you find the roots of the characteristic equation - part (b)?

Did you find the homogeneous solution of the ODE - part (c)?

What were those?
 
Are you familiar with Variation of Parameters?.

\(\displaystyle y''-5y'+6y=-3sin(2x)\)

We know the roots are 2 and 3, so we get for the auxiliary equation:

\(\displaystyle y_{1}=e^{2x},\;\ y_{2}=e^{3x}\)

\(\displaystyle y_{c}=C_{1}e^{2x}+C_{2}e^{3x}\)

The Wronskian:

\(\displaystyle W=\begin{vmatrix}e^{2x}&e^{3x}\\2e^{2x}&3e^{3x}\end{vmatrix}=e^{5x}\)

\(\displaystyle W_{1}=\begin{vmatrix}0&e^{3x}\\-3sin(2x)&3e^{3x}\end{vmatrix}=3e^{3x}sin(2x)\)

\(\displaystyle W_{2}=\begin{vmatrix}e^{2x}&0\\2e^{2x}&-3sin(2x)\end{vmatrix}=-3e^{2x}sin(2x)\)

\(\displaystyle u_{1}^{'}=\frac{W_{1}}{W}=3e^{-2x}sin(2x)\)

\(\displaystyle u_{2}^{'}=\frac{W_{2}}{W}=-3e^{-3x}sin(2x)\)

Integrate the u's:

\(\displaystyle u_{1}=\frac{-3}{4}e^{-2x}cos(2x)-\frac{3}{4}e^{-2x}sin(2x)\)

\(\displaystyle u_{2}=\frac{6}{13}e^{-2x}cos(2x)+\frac{9}{13}e^{-3x}sin(2x)\)

\(\displaystyle y_{p}=u_{1}e^{2x}+u_{2}e^{3x}=\frac{-15}{52}cos(2x)-\frac{3}{52}sin(2x)\)

The general solution is then:

\(\displaystyle \boxed{y=y_{c}+y_{p}=C_{1}e^{2x}+C_{2}e^{3x}-\frac{15}{52}cos(2x)-\frac{3}{52}sin(2x)}\)
 
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