z1(x) = 2x3 + x ln x, z2(x) = x ln x − x3 are solutions of a second order, linear nonhomo-geneous equation L[y] = f (x).y1(x) = x−2 is a solution of the corresponding reduced equation L[y] = 0. (a) Give a fundamental set of solutions of the reduced equation L[y] = 0. (b) Give the general solution of the nonhomogeneous equation L[y] = f (x).
3. Given that yı(t) = t, y(t) = t, and yz(t) = are solutions to the homogeneous differential equation corresponding to ty" + t'y" – 2ty' + 2y = 2*, t > 0, use variation of parameters to find its general solution.
4. Given that {cos x, sin 2, 1) is a fundamental set of solutions for y" + y = 0, solve the initial value problem with conditions y(0) = 3, y'(0) = 5, "(0) = -4. 4. Given that {coso, sin x 1} is a fundamental set of solutions for y'" + y = 0, solve the initial value problem with conditions y(0) - 3, V'(0) -- 5. "(0) -4
(8 pts) In this problem you will solve the non-homogeneous differential equation y" + 9y = sec (3x) (1) Let C and C2 be arbitrary constants. The general solution to the related homogeneous differential equation y" + 9y = 0 is the function yn (x) = C1 yı(2) + C2 y2(x) = C1 +C2 NOTE: The order in which you enter the answers is important; that is, Cif(x) + C2g(x) + C19(x) + C2 f(x). (2) The particular solution yp(x)...
Consider the differential equation y" – 7y + 12 y = 0. (a) Find r1, 72, roots of the characteristic polynomial of the equation above. 11,2 M (b) Find a set of real-valued fundamental solutions to the differential equation above. yı(t) M y2(t) M (C) Find the solution y of the the differential equation above that satisfies the initial conditions y(0) = -4, y'(0) = 1. g(t) = M Consider the differential equation y" – 64 +9y=0. (a) Find r1...
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It can be shown that yı = x-2, y2 = x-6 and y3 = 7 are solutions to the differential equation xạy" + 11xy" + 21y' = 0. W(y1, y2, y3) = For an IVP with initial conditions at x = 3, C1yı + C2y2 + c3y3 is the general solution for x on what interval? It can be shown that yı = x-2, y2 = x-7 and y3 = 5 are solutions to...
onsider the differential equation y" - 7y + 12 y = 3 cos(3t). (a) Find r. 12. roots of the characteristic polynomial of the equation above. ri, r2 = 3,4 (b) Find a set of real-valued fundamental solutions to the homogeneous differential equation corresponding to the one above. Yi (t) = 0 (31) »2(t) = 0 (41) (c) Find a particular solution y, of the differential equation above. y,(t) = Consider the differential equation y! -8y + 15 y =...
7. If y, and y2 are a fundamental set of solutions of t2y"-2/ + (3 + t)y = 0 and if won,n)(2) = 4, find the value of W(vi.v2)(3).
Please help with these two, thanks! In Exercises 1–17 find the general solution, given that y1 satisfies the complementary equation. As a byproduct, find a fundamental set of solutions of the complementary equation. 3. x2y'' − xy' + y = x; y1 = x 7. y''− 2y' + 2y = e^x*sec x; y1 = e^x cos x
1. (20 pts.) In the following Problems: (a) Seek power series solutions of the given differential equation about the given point xo ; find the recurrence relation. (b) Find the first four terms in each of two solutions yi and y2 (unless the series terminates sooner). (c) By evaluating the Wronskian W(y1, y2)(xo), show that yı and y2 form a fundamental set of solutions. (d) If possible, find the general term in each solution. i) y" +k+x+y = 0, 40...