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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.

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

## SOLUTION This solution is FREE courtesy of BrainMass!

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 and sine terms on both sides reveals that the coefficients of the cosine term must vanish (A=0), and:

Therefore

And the general solution is:

From initial conditions:

And:

Thus the solution to the IVP problem is:

And the derivative is:

The components graph is plotting y and y' vs. t
The orbit graph is plotting y' vs. y.

Both graphs are shown in the next page.

28.

Now we just repeat the same procedure as in the previous part, only now we solve for general ≠5.

The homogenous solution does not change:

And for the particular solution we guess:

Substituting this back into the equation we get:

Equating the coefficients of the cosine and sine on both sides of the equation we get:

So the particular solution is:

And the general solution is:

Using the initial conditions to solve for the constants we get:

So the general solution for this IVP is:

For =4 we just regain the equation of the previous part and its solution is:

And its solution curves and orbit curve already were plotted before.

For =1 the solution becomes:

And its derivative is:

The components and orbit graphs for this solution are plotted on the next page.

Together with the orbit plot for =4 we get a "cross-eyed heart":

Zooming in:

This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!