Help on a couple of problems

**ballajr** · March 24th 2010, 07:51 PM

1. a) Show that $t^r$ is a solution of Euler's equationL
$t^2y'' + aty' + by = 0, t > 0$
if $r^2 + (a-1)r + b = 0$

b) Suppose that $(a - 1)^2 = 4b$ . Using the method of reduction of order, show that $(lnt)t^((1-a)/2)$ is a second solution of Euler's equation.

2) Find the general solution of the equation.
$t^2(d^2y)/(dt^2) - t(dy)/(dt) + y = 0$

3) Use the Method of Variation Parameters to find the general solution of:
$(d^2y)/(dt^2) - 4(dy)/(dt) + 4y = te^{2t}$

4) Use the Method of Variation Parameters to solve the initial-value problem
$y'' + 4y' + 4y = t^{5/2}e^{-2t}; y(0) = y'(0) = 0$

5) Find the general solution to
$(tD - I)(tD + 3I)[y] = 32t^5 + 21t^4$
Hint: Use the idea of trying $y = t^r$ to find a particular solution.

**mr fantastic** · March 25th 2010, 01:36 AM

Originally Posted by ballajr

1. a) Show that $t^r$ is a solution of Euler's equationL
$t^2y'' + aty' + by = 0, t > 0$
if $r^2 + (a-1)r + b = 0$

b) Suppose that $(a - 1)^2 = 4b$ . Using the method of reduction of order, show that $(lnt)t^((1-a)/2)$ is a second solution of Euler's equation.

2) Find the general solution of the equation.
$t^2(d^2y)/(dt^2) - t(dy)/(dt) + y = 0$

3) Use the Method of Variation Parameters to find the general solution of:
$(d^2y)/(dt^2) - 4(dy)/(dt) + 4y = te^{2t}$

4) Use the Method of Variation Parameters to solve the initial-value problem
$y'' + 4y' + 4y = t^{5/2}e^{-2t}; y(0) = y'(0) = 0$

5) Find the general solution to
$(tD - I)(tD + 3I)[y] = 32t^5 + 21t^4$
Hint: Use the idea of trying $y = t^r$ to find a particular solution.

Please don't post more than two questions in a thread. Otherwise the thread can get convoluted and difficult to follow. It is better for forum organization and better for you to get your questions answered in a more timely manner if you start new threads as necessary for remaining questions. eg. If you have five questions, post two of them in two threads and start a new thread for the remaining one etc.

And if the question has more than two parts to it, it is best to post only that question and its parts in the thread and start a new thread for other questions.

By the way, what attempt have you made? eg. Are you honestly stuck with Q1 a?

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