As a specific example we consider the non-homogeneous problem y′′−16y=48e8x (1) The general solution of the homogeneous problem (called the complementary solution, yc=ay1+by2yc=ay1+by2 ) is given in terms of a pair of linearly independent solutions, y1y1, y2y2. Here aa and bb are arbitrary constants. Find a fundamental set for y″−16y=0y″−16y=0 and enter your results as a comma separated list
∗∗BEWARE Notice that the above set does not require you to decide which function is to be called y1y1 or y2y2 and normally the order you name them is irrelevant. But for the method of variation of parameters an order must be chosen and you need to stick to that order. In order to more easily allow We B Work to grade your work I have selected a particular order for y1y1 and y2y2. In order to ascertain the order you need to use please enter a choice for y1=y1= and if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box marked as correct then y2=y2= .With this appropriate order we are now ready to apply the method of variation of parameters.
(2) For our particular problem we have W(x)=W(x)=
u1=∫−y2(x)f(x)W(x)dx=∫u1=∫−y2(x)f(x )W(x)dx=∫ dx=dx= u2=∫y1(x)f(x)W(x)dx=∫u2=∫y1(x)f(x)W (x)dx=∫ dx=dx=
And combining these results we arrive at
yp=yp=
(3) Finally, the general solution is y=yc+ypy=yc+yp where yc=ay1+by2yc=ay1+by2 where aa and bb are arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions y(0)=2y(0)=2 and y′(0)=4y′(0)=4.