Question

help solving the unique solution of the IVP

W *) As a specific example we consider the non-homogeneous problem y" - 16y = 482** (1) The general solution of the homogeneous problem (called the complementary solution, y Independent solutions, y. y. Here a and bare arbitrary constants. d y + by is given in terms of a pair of linearly Find a fundamental set for y' - Toy = 0 and enter your results as a comma separated list 6^(4x),0^-(4x) BEWARE Notice that the above set does not require you to decide which function is to be called y ory 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 WebWork to grade your work I have selected a particular order for y, and y. In order to ascertain the order you need to use please enter a choice for = (4x) 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 y = 0^{-4x) . 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) = -8 2 dx = (-4-4x)(48e^(8x11-8 dx = 3/2 (4x) W(x) y(x)(x) dx = W(x) e^(4x)(48e (8x)V-8 dx = -1/2 (12x) And combining these results we arrive at y = e^(4x)(3/2e^(4x))+e^(-4x)(-1/2e^(12x)) (3) Finally, the general solution is y = y + y, where yc = a + by2 where a and bare arbitrary constants. Use the general solution to find the unique solution of the IVP with initial conditions (0) = -2 and y(0) = 12. y = (.5)^(4x)+(-2.5)(-4x)+(e (4x)(3/2e^(4x))+eW-4x)(-1/2 (12x})

Answer 1

Given DiFfemtial equation is 16y 4e Homogeneo diffeint i'a equatan cansider y" - 163 = O Solution of these homo. eq 3 6f the forry Shere a Sb are bitort constant consider aurilary ean af m2-1=0 . m 14 -4X ae2 be . 42 ond use variatlon ot parameder +o olve Now H eTe Ч1 e4x e -4 ce) x (ett) a - 4-4 92x) fx) dx w (2) NOCo 82 -4X x 48e e dz - 8

dx 4x e ニ (xh e4x 3 2 Noo w x) 4x e人48e -8 12 2 6e XP 12 (4 -6 1 2 yp IuI+Y242 4 x e - 42 e =e Jemau Jo i .. yetyp and Rom e e tge +be x 4se yco- -2 3 - 2 = +a +b > 「+a+b ニ-2

Nous differntate (8)+ 4a4be use KV-4 +4a-4b 4a-4b-4 a-b- Fiom a tb =-3 ab 2a -2 a=-I and put a- -1-b= 1 -b = i+ b-2 9enn Rom 5 le - R e yoa) --) - e4r (1 +2) - 24 -3e e