Question

(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de and the W() arbitrary constant of integration to be zero In other words we have the single integral formula 1 (2) - (2) vi(I)f(I) de 1(c)f(T) de +uta) { "WIE WE) As a specific example we consider the non-homogeneous problem y+by' + 8y - 4 sin(e) (1) The general solution of the homogeneous problem called the complementary solution, ay, + by its given in terms of a pair of linearly independent solutions. With Here a and bare arbitrary constants Find a fundamental set for y" + y + 8y = 0 and enter your results as a comma separated list 41-41 BEWARE Notice that the above sel does not require you to decide which function is to be called y Of V 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 Work lo grade your work I have selected a particular order for Vi and y. In order to ascertain the order you need to use please enter a choice for and if your answer is marked as incorrect simply enter the other function from the complementary set. Once you get this box manded as correct then With this appropriate order we are now ready to apply the method of variation of paramelers (2) For our particular problem we have W(z) 4(cos(4x)+sin 2(4x)) (2)(2) W(z) (2)/(z) dar ty W And combining these results we arrive at Wy (3) Finally, using a and b for the arbitrary constants in We, the general solution can then be written as yW

Answer 1

y" +64 +8y= 48in (622, The Ae corresponding to get sy + 8y =0 (8 + 6 + 277=0 is given by m² + 6 m+ 8 =0 mt4meam +8:07 mln+4) + (+4)=0 me 200 m2 450 m.-286-4. Hence C.f. = y c = gege so the fundament of sel ts Le é an and e 4x 1 Now dakce -4x then wronscian who .62 =-4e 6-6x tae 201 -é verbal Now 4 1 tawfix) -du -2.86% Inicy=-226% dx = stety 48ince 2x 2 lead sin le lo (where fous = 4linke? My dw] - con reoxy and was confinseng 4, Since) dx = - 282283.impetitie = con lear Hence & pla) = yras 4, c) * Yalx). Ugn) = at - concear, f + 42. concey Joena)= le **e XJ ascens Ten the goveral solution of he given different al equation in yeas= y(y (8) yin= qe at CE A LEAL ex costeak Anner