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**ballajr** 1. a) Show that $\displaystyle t^r$ is a solution of Euler's equationL

$\displaystyle t^2y'' + aty' + by = 0, t > 0$

if $\displaystyle r^2 + (a-1)r + b = 0$

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

2) Find the general solution of the equation.

$\displaystyle t^2(d^2y)/(dt^2) - t(dy)/(dt) + y = 0$

3) Use the Method of Variation Parameters to find the general solution of:

$\displaystyle (d^2y)/(dt^2) - 4(dy)/(dt) + 4y = te^{2t}$

4) Use the Method of Variation Parameters to solve the initial-value problem

$\displaystyle y'' + 4y' + 4y = t^{5/2}e^{-2t}; y(0) = y'(0) = 0$

5) Find the general solution to

$\displaystyle (tD - I)(tD + 3I)[y] = 32t^5 + 21t^4$

Hint: Use the idea of trying $\displaystyle y = t^r$to find a particular solution.