The University of Sydney School of Mathematics and Statistics Assignment 1 MATH2065: Introduction to PDEs Semester 2, 2017 Web Page: http://http://www.maths.usyd.edu.au/u/UG/IM/MATH2065/ Lecturer: F Viera Due Thursday 7 September (Week 6) by 5:00pm via Turnitin. A single PDF file must be uploaded to Turnitin through the Learning Management System (Blackboard). ZIP files are not accepted by Turnitin. Neat handwritten solutions are acceptable; however, illegible work will not be marked. Please, make sure you receive a Turnitin Digital Receipt confirming that you have successfully uploaded your assignment. Save this receipt. 1. Use the standard method (not Laplace transforms) to find the solution y(t) of the ordinary differential equation y 00 + 4 2 y = 3 2 sin t, that satisfies the initial conditions y(0) = 0 and y 0 (0) = 0, where > 0 is a given real constant. 2. Use the method of Laplace transforms to solve the following differential equation for y(t): d2 y 9y = e3t , 2 dt that satisfies the initial conditions y(0) = 0 and y 0 (0) = 0. 3. Consider the following partial differential equation for u(x, t) \u0012 \u0013 2 u 2u f (x) = x2 x t2 If f (x) is a function of x only, but not of t, derive two ODEs by separation of variables. (You do not need to solve these ODEs.) c 2017 The University of Sydney Copyright