MATH300 Lecture 21: 21. Separation of Variables, Laplace on a Disk

4. Use Laplace transforms to solve the following differential equation 5. Consider the following list of differential equations. Do not attempt to solve any of them. Rather, identify which solution technique would be appropriate to either approximate the solution or solve the equation exactly. The solution techniques are: Euler's method (before or after rewriting the equation as a syste), phase line analysis, phase plane analysis (before or after rewriting as a system), reverse product rule, separation of variables, Laplace transform (a) x' + z = sin(t): x(0) = 1 (b) r" +x = sin(t); r(0) = 1: z'(0) =-1 (e) z, = r(1-z); x(0) = 2 Briefly defend your answers

the table table and properties of Laplace transforms provided on the back page to Usenne the following Laplace transforms: (10 pts each) (a) S'{f'( cost. t)) where f (t)e a. (b) 묘{(1 + e-t)2 + 4t3e2t}

5. Consider the following list of differential equations. Do not attempt to solve any of them. Rather, identify which solution technique would be appropriate to either approximate the solution or solve the equation exactly The solution techniques are Euler's method (before or after rewriting the equation as a system), phase line analysis, phase plane analysis (before or after rewriting as a system), reverse product rule, separation of variables, Laplace transform (a) z' + z = sin(t): r(0) = 1 (b) r" +z = sin(t); r(0) = 1: r,(0) =-1 (e) z' = r(1-z); x(0) = 2 Briefly defend your answers.