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# Cross-eyed heart

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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.
Any surprises?

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.

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https://brainmass.com/math/graphs-and-functions/differential-equations-cross-eyed-heart-28848

#### Solution Preview

Hi there!

Very cute!

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 ...

#### Solution Summary

The solution shows how to solve the required differential equations and provides the graphs that show an interesting behavior. The plot behavior is analyzed.

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