Download Exam 2 Review Sheet for Differential Equations | Math 315 and more Exams Differential Equations in PDF only on Docsity! Math 315-1 Exam 2 Review Sheet Exam number three covers sections 3.4, 3.5, 3.7, 4.1, 4.2, 10.1, and what we covered in chapter 5. No graphing/symbolic calculators allowed, but a 3” by 5” note card is allowed. Chapter 3: Second Order Differential Equations 4. Repeated Roots; Reduction of Order: Be able to solve a second order differ- ential equation where the characteristic equation gives repeated roots. Be able to solve an associated initial value problem. Be able to use the method of reduction of order to find the second solution of a differential equation given a solution. 5. Nonhomogeneous Equations; Method of Undetermined Coefficients: Given a nonhomogeneous differential equation, be able to find the homogeneous and particular solutions in order to construct the general solution. Be prepared for a nonhomogeneous term that is an exponential, sin or cos, or a polynomial. Also be prepared for a nonhomogeneous term that is the product and/or sum of any of these. What happens if the nonhomogeneous part is part of the homogeneous solution? 7. Mechanical and Electric Vibrations: What is an undamped free vibration, what is critical damping, and what is overdamped? Chapter 4: High Order Linear Equations 1. General Theory of nth Order Linear Equations: Be familiar with an nth order differential equation. Know when and under what conditions we have exactly one solution. How does the solution to an initial value problem, linear independence of solutions y1, y2, ·, yn, and the Wronskian relate? If the equation is nonhomogeneous, what do we add together to find the general solution? 2. Homogeneous Equations with Constant Coefficients: Be able to solve a differ- ential equation of this form. Note that the roots of the corresponding characteristic equation can be all real and distinct, be complex conjugate pairs, or be real and repeated. Be prepared for any of these situations. Remember how to factor. The rational root theorem may help. 1
