27. y'' + 25y = sin(4t),
y(0) = 0, y'(0) = 0.
Plot the component curves and the orbit, the latter for the rectangle |y|< 0.25 and |y'|< 1.
28. Hearts and Eyes: Find a solution formula for y'' + 25y = sin&(wt), where w is not equal 5. Plot the solution curve of the IVP with y(0) = y'(0), where w=4. Plot the orbit for 0< t <20 in the rectangle |y| < 0.1, -0.5 < |y'| < 0.3.
Repeat with w=1. Overlay the graphs.
Attached is the solution with the required graphs.
First let's solve the equation:
We start by first solving the homogenous equation:
Which has a characteristic equation:
So the homogenous solution is:
Since the independent term is not part of the homogenous solution, we guess a solution of the form:
Substituting it back in the equation we get:
Equating the coefficients of the cosine ...
The solution shows how to solve the required differential equations and provides the graphs that show an interesting behavior. The plot behavior is analyzed.