110425  Differential Equations

 1. The equation y ' = A * y, for A = [0, 2; -2, 0] is found to have solution set {[cos(2 t); -sin(2 t) ], [-sin(2 t); -cos(2 t)]}

Write the equation as a system of two equations in terms of the variables y_1 and y_2.

Give two linearly independent solutions y_1 and y_2 for this system, based on the solution set as given above.

Give the fundamental set, based on your solutions y_1 and y_2.

Show that your fundamental set is really a fundamental set. If you have correctly based your set on the solution set given above, it will be so.

2. For the equation given in the preceding, y_1 and y_2 were given a meaning in terms of the system of equations.

Now we will consider y_1 to be the vector [cos(2 t); -sin(2 t) ], y_2 to be the vector [-sin(2 t); -cos(2 t)].

write the matrix psi(t) = [y_1, y_2]

Show that this matrix solves the equation

psi ' (t) = A * psi(t).

3. If y_P = psi(t) u, where u = [u_1, u_2] and psi(t) is a solution matrix for the system y ' = A y, then

what is the expression for y_P '?

If we substitute y_P ' into the equation

y ' = A y + g,

then what do we get?

Considering that y ' = A y, how does our equation simplify?

How would we use this to solve for the vector u?

Having found the vector u, how do we then find y_P?

4. Apply the above to the equation y ' = A y + [e^t; 2], with A = [0, 2; -2, 0 ].

5. In 2-dimensional real space, let B1 = {v_1, v_2} = { [1; 0], [0; 1] } be the standard basis.

Let B2 = {u_1, u_2} = { [1, 1], [1, -1] } be another basis.

Express the vector [2; 3] in the form c_1 v_1 + c_2 v_2.

What therefore is the vector [c_1; c_2]_B1 that expresses [ 2; 3] in terms of the basis B1?

Now express the same vector in the form d_1 u_1 + d_2 u_2.

What therefore is the vector [d_1, d_2]_B2 that expresses [2, 3] in terms of the basis B2?

If

u_1 = a_11 v_1 + a_12 v_2,

and

u_2 = a_21 v_1 + a_22 v_2,

then what are a_11, a_12, a_21 and a_22?

Show that if T is the 'transformation matrix'

T = [a_11, a_12; a_21, a_22]

then

T * [c_1; c_2] = [d_1; d_2].

... generalize ...