Math 555 Differential Equations I

Unit II Exam: Review Guide

1. Determine whether or not the functions are linearly independent.

	a.)	\(\left\{e^t,te^t\right\}\)
	b.)	\(\left\{ x^2, x\vert x\vert \right\}\)
	c.)	\(\left\{ \sin t, \cos t \right\}\)
	d.)	\(\left\{ t, te^t, t^2e^t \right\}\)

2. Find the general solutions of the homogeneous equations.

	a.)	\( y'' + 8y' - 9y = 0 \)
	b.)	\( y'' - 2y' + 6y = 0 \)
	c.)	\( 25y'' - 20y' + 4y = 0 \)

3. Consider the differential equation \((x-1)y'' - xy' + y = 0, x > 1\). Given that \(y_1(x) = e^x\) is a solution, use the method of reduction of order to find the general solution of the DE.

4. Find the general solution of the differential equation \( t^2y'' - 4ty' + 4y = 0, t > 0\). [Hint: assume the general form of the solution is \(\varphi = t^r \).]

5. Find the particular solution of the initial value problem. \[\begin{cases} y'' - 2y' + y = te^t + 4, \\ y(0) = 1, \\ y'(0) = 1. \end{cases}\]

6. Find the general solution of the differential equation \(y'' + 9y = 9\sec^2(3t)\).

7. You wish to use the method of undetermined coefficients to solve the differential equation \[ y^{(4)} + 2y'' + y = 3 + \cos(2t) + 4e^{-t}.\] Write the form of the guess \(Y(t)\) for the non-homogeneous part. Do NOT solve for the unknown coefficients.

8. You wish to use the method of variation of parameters to solve the differential equation \[ y''' - y' = \csc t.\] Write the integral form of the solution, but do not compute the integrals. (... unless you really want to. They aren't that bad.)

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