Use Mathematica to find the general solution to the following differential equations,
and the specified family of solutions. Then plot simultaneously this family of solutions.

Computer Lab Work.

Exercise 1. Find the solutions to the D. E.
with the initial conditions y[0] = 1, and for m = -10, -8, -6, ... , 6, 8, 10.
Plot simultaneously this family of solutions.

Exercise 2. Find the solutions to the D. E.
with the initial conditions y[0] = for c = 0, 1, ... , 8, 9, 10 and .
Plot simultaneously this family of solutions.

Exercise 3. Find the solutions to the D. E.
with the initial conditions y[0] = 1, and for m = -5, -4, -3, ... , 3, 4, 5.
Plot simultaneously this family of solutions.

Exercise 4. Find the solutions to the D. E.
with the initial conditions y[0] = for c = 0, 1, ... , 8, 9, 10 and .
Plot simultaneously this family of solutions.

Exercise 5. Find the solutions to the D. E.
with the initial conditions y[0] = 1, and for m = -5, -4, -3, ... , 3, 4, 5.
Plot simultaneously this family of solutions.

Exercise 6. Find the solutions to the D. E.
with the initial conditions y[0] = for c = 0, 1, ... , 8, 9, 10 and .
Plot simultaneously this family of solutions.

Exercise 7. Find the general solution to the D. E. .

Find the roots of the characteristic equation.

Form the general solution using the roots of the characteristic equation.

Verify that f[x] and its derivatives satisfy the D. E.

Exercise 8. Find the general solution to the D. E.
.

Find the roots of the characteristic equation.

Form the general solution using the roots of the characteristic equation.

Verify that f[x] and its derivatives satisfy the D. E.

Remark the general solution using real functions.
This can be accomplished by considering the complex pair.

Look at their real and imaginary parts.

The two linearly independent real functions are:

Use them in forming a new general solution.

Verify that f[x] and its derivatives satisfy the D. E.

Solutions.

Return to the Differential Equations Project

Return to the Numerical Analysis Project

Return to the Complex Analysis Project

(c) John H. Mathews, 1998