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EXERCISE 1. In each of Problem (1) through (2) solve the initial-value problem 2.y/#-y=0, EXERCISE 2. In each of Problem through 3. find the general solution of the given differential equation 3. y" + where X>0 and A# 77277 for m = 1,... N. EXERCISE 3. Answers each question of Problems through 6. 1. Show that the Wronskian W(cos20; - cos 20) is zero. 2. If f,g and h are differentiable functions, show that W(fg: fh) PW(g:h). 3. fthe Wronskian W(f;g)of f and g, and ifu=2f-g,v=f +2g, find the Wronskian W(u;v) vin terms of w(f:g). 1 4. If the Wronskian W(f,g) of / and gis toost sint, and /+ 3g. f g. find the Wronskian W(u,v) of and v. 5. If the Wronskian w(its g) of f and gis t²e², and f(t) t, find g(t). 6. Verify that the functions y1 and y/2 are solutions of the given differential equation. Do they constitute a fundamental set of solutions? Justify your answer. xy +y=0, 0EXERCISE 4. Let p,g and a be continuous functions on an open interval l,and 4/1 and Y/2 fundamental set of solutions of the homogeneous equation corresponding to the nonhomogeneous equation y/" (1) It is known from Theorem 3.6.1 Textbook, Page 189-190 ) that particular solution of Eq.(1) Y(t) -yr(t) i W(41,32)(s) y2(s)g(s) ds y2(t) W y1(s)g(s) ds. (2) (y1 ,y2((s) where to is any conveniently chosen point in I. 1. Show that Y(t) becomes to 2. Use Lemma to show that Y(E) is solution of the initial value problem 35" ppt)y' q(t)y g(t), 3. Use the results of questions 1. and 2. to show that the solution of initial value problem is Lemma: Let f(x,4), a(t). B(t) be smooth functions Let g(E) Then, g(£) is differentiable and g²(t) f(3(t), ) f(a(t),t) d(t). 2 EXERCISE 5. 1. Let y and !/2 be solutions of the differential equation y" + g(t)y =0, where p and are continuous on an open interval. Show that the Wronskian W y2)(t) is given by W(y1.32)(t) cexp [-fruca) where c is certain constant that depends on y1 and 3/2 but not on t 2. If g1 is known nonvanishing solution of y" p(t)y g(t)y =0. show that second solution satisfies (m) W(y/w,ye) yi where W(Y) ye) is a the Wronskian of y and 92- 3. Use the formulas of questions 1. and 2. above to find a second independent solution of the given differential equation 4. By using the results of Theorem 3.6.1 (Textbook, Page 189-190), verify that the given functions V1 and Y/2 satisfy the corresponding homogeneous equation; find particular solution and the general solution of the given nonhomogeneous equation 4y x²Inz,

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