School of Mathematical Sciences Engineering Mathematics IIA, MATHS 2201 Assignment 6 question sheet Due: Tuesday, 02/05/2017 (Week 8), by 12. It is…

Show that(3×2 + y cos x) dx + (sin x ???? sin y) dy = 0;is exact. Hence, or otherwise, solve the initial value problem(3×2 + y cos x) dx + (sin x ???? sin y) dy = 0; y() =2:[8 marks]2. Verify that y1(x) = e????x is a solution of(x + 1)y00 + xy0 ???? y = 0; x > ????1: (1)Use reduction of order to nd another linearly independent solution and hence writedown the general solution. [12 marks]3. Consider the problem of an undamped mechanical system upon which an external forceF acts. Suppose the external force acts only for a nite interval of time, 0  t  .Then, the mechanical system can be described by a mathematical model in which thedisplacement y(t) satises the dierential equationy00 + 4y =(F(t); 0  t  ;0; t > :(2)Assume that the applied force is given byF(t) = sin 2t for 0  t  .(a) Using the method of undetermined coecients, nd the general solution of thenonhomogeneous dierential equation,y00 + 4y = sin 2t for 0  t  .(b) If the system is initially at rest, determine the values of y and y0 at t = .(c) Write down the general solution of the homogeneous dierential equation,y00 + 4y = 0 for t > .(d) In this model, it is natural to insist that the solution to (2) (the displacementy) and its rst derivative (the velocity y0) are continuous at t = . Using thisfact, and your result from part (b), determine the displacement and velocity of thesystem at time t = 2.[18 marks]

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