Assignment 5 Due: Fri. Mar. 31, 2000

1. Given that x = t2 is a solution of the differential equation

tx´´ + (t-1)x´ - 2x = 0,

find a second linearly independent solution expressed as an integral.

2. Find the general solutions to the following Cauchy-Euler differential equations:

3. Find the general solution to the following differential equations:

(i) x´´ + x = cot(2t) [Hint: express cot(2t) in terms of cos(t) and sin(t)].

(ii) x2y´´ - xy´ + y = x ln x [Note that here primes indicate differentiation with respect to x].

4. Use the Wronskian to show that the following pairs of functions are linearly independent:

(i) t2, t3

(ii) t, e2t

(iii) sin(t), sin(2t)

5. (a) Solve the boundary-value problem

(b) For what values of the parameter q > 1 does the following boundary-value problem have a non-trivial solution?

x´´ + 2x´ + qx = 0; x(0) = 0, x(1) = 0

6. Find the general solution of the following differential equations as a power series about x0 = 0.

(i) y´´ - 2xy´ + y = 0

(ii) (x2 + 1)y´´ + xy´ + 3y = 0.

Find the recurrence relation for the coefficients, write out the first four terms of each series and, if possible, find a general expression for the coefficients.

7. What would the solution to the differential equation in question 6 (ii) be if it satisfied the initial conditions y(0) = 2, y´(0) = -2?

8. Use the method of Frobenius to solve the following differential equations

(i) 3xy´´ + y´ - y = 0

(ii) 4x2y´´ - 4xy´ + (3 - 4x2)y = 0

Note that in part (ii) the roots of the indicial equation differ by an integer but nevertheless a separate solution can be obtained for each root.

9. Use the method of Frobenius to find one solution of the following differential equations and write down the form of the second solution.

(i) x2y´´ -xy´ + (1-x)y = 0

(ii) xy´´ + 4y´ - xy = 0

10. Use MAPLE to plot solutions of x´´+ 100x = 20cos(wt) for values of w = 8, 9 and 10. Note the behaviour of the solutions. Take x(0) = 0 and x´(0) = 0, use a stepsize parameter of 0.01 and plot the results for t in the interval from 0 to 10.

11. We can 'tune' an RLC circuit by changing the capacitance C to select a certain frequency. Consider such a circuit with L = 1, R = 0.1 and E = cos(t) + cos(5t) and initial conditions Q(0) = 0 and I(0) = 0. Use MAPLE to plot the current in the circuit for values of C = 1, 1/25 and 1/81 and t in the interval from 70 to 100. Note the amplitude of each of the solutions. Also plot each term in E separately and compare the frequencies with the solutions for the current you have obtained.