Math 555 Differential Equations I |
Group I: There will be one question from this section on the makeup exam.
1. Solve the equation \( (2y + x^2y)y' = x\).
2. Solve the equation \( (2y+1)dx + (x - \tfrac{y}{x})dy = 0 \).
3. Find the solution to the initial value \(y' = 2xy^2\), \( y(0) = -1\),
and determine the domain of the solution.
Group II: There will be one question from this section on the makeup exam.
4. Solve the initial value problem \(y'' + 4y' + 3y = 0\), \(y(0) = 2\),
\(y'(0) = -1\).
5. Consider the equation \( xy'' + (3 -2x)y' - 4y = 0\), \(x > 0\). If
\(y_1(x) = x^{-2}\) is a solution of the equation, use the method of reduction of
order to find the general solution.
6. Solve the initial value problem.
\[
\begin{cases}
y'' - 2y' + 5y = 8\sin x - 4\cos x, \\
y(0) = 3,\ \ \ y'(0) = 9.
\end{cases}
\]
Group III: There will be one question from this section on the makeup exam.
7. Consider the equation \(y'' - 4y' + 4y = x^2e^x + 4xe^x\sin x
+ 6\cos x + 3\). Determine a suitable form of a particular solution by the
method of undetermined coefficients. Do not evaluate the coefficients.
8. It is known that \(x^2\) and \(x^2\ln x\) are solutions of the
homogeneous equation associated to the nonhomogeneous equation
\[ x^2y'' -3xy' + 4y = x^2\ln x. \]
Use the method of variation of parameters to find a particular solution to the
nonhomogeneous equation.
9. Use the power series method to find a fundamental set for the equation
\(y'' - 3xy' + y = 0\). Determine the first three terms in each of the two
solutions that form the fundamental set.
Group IV: There will be one question from this section on the makeup exam.
10. Given the equation \(x^2y'' + 8xy' + 12(1+x)y = 0\). Then \(x = 0\)
is a regular singular point. Show a correct form of the series solutions to the
equation.
11. Find the inverse Laplace transform for \(F(s) = \dfrac{e^{-2s}(s-1)}
{s^2+2s+5}\).
12. Use the Laplace transform to solve the problem.
\[\begin{cases}
y'' + 2y' + 5y = 0 \\
y(0) = 2,\ \ \ y'(0) = -1
\end{cases}\]
13. Use the Laplace transform to solve the problem.
\[\begin{cases}
y'' + 4y + u_\pi (t) \\
y(0) = y'(0) = 0.
\end{cases}\]
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