EE 518 Homework 2 Due on Monday September 12 2016 Problem 1 Solve the initial value problem x dy y x2 sin x dx y 0 dy We first write the equation in the standard form dx p x y q x dy y dy 2 x dx y x sin x dx x x sin x Thus p x x1 q x x sin x R R 1 Hence the integrating factor is I x e p x dx e x dx e ln x x1 Therefore R R R R y x e p x dx q x e p x dx dx c x x sin x x1 c x cos x c Since y 0 we can calculate c cos c 0 c cos 1 Thus y x x cos x 1 x cos x x Problem 2 Find the general solution for the given Differential Equations i dy y e2x dx ii y 0 2xy 2x3 dy 1 The equation is already in the standard form dx p x y q x dy 2x y e dx Thus p x 1 q x e2x R R Hence the integrating factor is I x e p x dx e 1dx e x Therefore R R R R R y x e p x dx q x e p x dx dx c ex e2x e x dx c ex ex dx c ex ex dx c dy p x y q x 2 The equation is already in the standard form dx 0 3 y 2xy 2x Thus p x 2x q x 2x3 R R 2 Hence the integrating factor is I x e p x dx e 2xdx ex Then we multiply both sides with I x and so we have that R R R R R 2 2 2 2 y x e p x dx q x e p x dx dx c e x 2x3 ex dx c e x 2xex x2 dx c R 2 2 2 2 2 2 2 e x ex x2 ex 2xdx c e x ex x2 ex c x2 1 ce x Problem 3 Find the general solution for the given Differential Equations i y 000 y 00 y 0 y 0 Page 1 of 4 EE 518 Homework 2 Due on Monday September 12 2016 ii D 2 D 3 y 7e2x 1 From the given equation we get D3 D2 D 1 0 D 1 D2 1 0 D 1 D i D i This means that we have the linearly independent solutions ex eix e ix or ex cos x sin x These sets give as the solutions y x c1 ex c2 eix c3 e ix or y x c1 ex c2 cos x c3 sin x 2 From the given equation we get D 3 D 2 This means that we have the linearly independent solutions e3x e2x These sets give as the solutions yc x c1 e3x c2 e2x Since the given equation is of the form P D y F x Ae x we have that y x yc x yp x c1 e3x c2 e2x A0 xe2x Then D2 yp 5Dyp 6yp D2 A0 xe2x 5D A0 xe2x 6 A0 xe2x 7e2x A0 7 Thus the solution is y x c1 e3x c2 e2x 7xe2x Problem 4 Suppose y1 x e 2x xe x y2 x xe 2x xe x and y3 x e 2x xe 2x xe x are three special solutions to the linear differential equation y 00 1 y 0 2 y F x where 1 and 2 are constants 1 Find 1 2 and F x 2 Determine the general solution to the linear differential equation 3 If f x is a special solution to the linear differential equation and f 0 0 f 0 0 0 then find f x 1 Let L D2 1 D 2 Then Ly1 x F x Ly2 x F x Ly3 x F x since they are special solutions Thus L y3 x y1 x L y3 x y2 x 0 This means that y3 x y1 x xe 2x and y3 x y2 x e 2x are two special solutions of the associated homogeneous equation y 00 1 y 0 2 y 0 Since xe 2x and e 2x are linearly independent we have that L D 2 2 which gives us 1 4 2 4 xe x is a special solution to Ly F x so we have that F x xe x 00 4 xe x 0 4 xe x x 2 e x 2 The general solution can be written as y x c1 e 2x c2 xe 2x xe x where c1 c2 3 Let f x c1 e 2x c2 xe 2x xe x Then f 0 0 c1 0 and f 0 0 0 c2 1 0 c2 1 Thus f x xe 2x xe x Page 2 of 4 EE 518 Homework 2 Due on Monday September 12 2016 Problem 5 This is a Matlab exercise Produce a figure with two different subplots In the first subplot plot the functions y x sin x z x cos x 4 w x sin x cos x8 for x 0 10 and in the second subplot plot the function z x y sin x2 y 2 x2 y 2 for x y 10 10 Put titles label your axes and put grids For the 2D plot customize all the lines change the colors and put a legend For this exercise please provide your script along with the plots that you produced printed Page 3 of 4 EE 518 Homework 2 Sample code and Due on Monday September 12 2016 plots Page 4 of 4