User:EGM6321.f12.team4.Rui.C/Homework7 R7.2

Statement
Consider the L2-ODE-CC:

Initial conditions:

Use variation of parameters to show that:

Solution
In order to obtain at least the 1st homogeneous solution, we use trial solution:

and substitute it into homogeneous version of (7.2.1):

Here the exponential term $$e^{\lambda t}$$ can be canceled and then we derive the characteristic equation:

The two characteristic roots are

It is obvious that we have obtained both the 1st and the 2nd homogeneous solution, because both of the characteristic roots are constant. We then choose $$ y_{1} = e^{i \sqrt{a_0}t}$$ as the 1st homogeneous solution and use variation of parameters by assuming:

By substituting it into (7.2.1):

Assume Z:

By IFM, we assume $$h=h(t)$$, then

Then

Then U can be found to be:

Thus y(x) can be obtained:

Substituting expression for h, (7.2.13) into the above expression:

The first term turns out to be in a form of Fourier transformation where the function to be transformed is $$e^{-i\sqrt{a_0}s}f(s)$$. And through the convolution theorem， it can be derived as:

Thus the solution can be written as:

If we only consider real part of the solution, the expression (7.2.21) can be reduced to the form:

To determine the two constants, we plug the solution into two initial conditions:

We can then solve $$C_1',C_2$$ in terms of $$y(t_0),y'(t_0)$$ :

By substituting these two coefficients back into the expression (7.2.22) and reorganization, we can derive the final form of the solution:

(7.2.26) is the required expression and we note that it should be $$\sqrt{a_0}$$ instead of $${a_0}$$ in the solution. This is also reasonable in the perspective of unit, because in a spring-mass system y(t) has a unit of length, $$\sqrt{a_0}$$ has a unit of frequency, and y'(t) has a unit of velocity. Then $$\frac{y'(t)}{\sqrt{a_0}}$$ will have a unit of displacement, too.

Author and References

 * Solved and Typed by -- Rui Che