Given yı(x) = x4 satisfies the corresponding homogeneous equation of x+y" + 3xy' – 24y =...
(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)...
Find a particular solution, yp(x), of the non-homogeneous differential equation d2 +y(x) = 6 ((x)) +9 y(x) = 6 x+2, d x2 given that yh(x) = A e3x +B x @3x is the general solution of the corresponding homogeneous ODE. The form of yp(x) that you would try is Oyp = ax + b Oyp = a 2x Oyp = ax2 3x Enter your answer in Maple syntax only the function defining yp(x) in the box below. For example, if...
4. given that yı = is a solution of the homogeneous equation. (1 + x2)" + 4xy' + 2y = 0 (a) Find y2 using the reduction of order formula. 7 pts (b) Use Wronskian to verify that yi and Y2 are linearly independent solutions. 5 pts
Verify that the given functions Y1 and y2 satisfy the corresponding homogeneous equation; then find a particular solution of the given nonhomogeneous equation. x2y" – 3xy' + 4y = 7x? In x, x>0; 71(x) = x2, yz(x) = x2 In x Y(x) =
A.9. First-order linear non-homogeneous ODEs having one dependent variable are of the form dy + P(x)y = f(x). Beginning with yp = uyż, where yı = e-SP(x)dx and is thus a solution to Y + P(x)y = 0, and given that the general solution y = cyı + Yp, use variation of parameters to derive the formula for the general solution to first-order linear non-homogeneous ODES: dx y = e-SP(x)dx (S eS P(x)dx f(x)dx + c). You may use the...
(b) (30 points) Solve the following IVP given that yı (x) = x and y2 (x) = x3 are solutions to the corresponding homogeneous differential equation. Be sure to fully evaluate any integrals that arise. (You will have to use polynomial long division.) " - 3xy' + 3y = y(1) = 0, y(1) = 0 1+2
Given a second order linear homogeneous differential equation a2(x)” + a (x2y + a)(x2y = 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, y. But there are times when only one function, call it yi, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the az(x) + 0 we rewrite...
Differential Equations for Engineers II Page 2 of 6 2. Consider the nonhomogeneous ordinary differential equation XY" + 2(x – B)y' + (x – 2B)y = e-1, x > 0, (2) 5 marks where ß > 0 is a given constant. (a) A solution of the associated homogeneous equation is yı = e-*. Use the formula for the method of reduction of order, as described in the lecture notes / record- ings, to find a second solution, y2, of the...
(1 point) In this exercise you will solve the initial value problem e-9 y" – 184' +81y = 4472; y(0) = -3, v'(0) = -2. (1) Let C and Cybe arbitrary constants. The general solution to the related homogeneous differential equation y" – 18y' +81y = 0 is the function yh() = C1 yı() + C2 y2() = C1 +C2 NOTE: The order in which you enter the answers is important; that is, Cif(T) + C29(2) #C19() +C2f(). is of...
Solve the differential equation below using series methods. y' + 3xy' + 8y = 0, y(0) -1, y'(0) = – 5 Find the first few terms of the solution y(x) = axxk. k=0 ao = Preview ai Preview A2 = Preview = a3 = Preview 24 = Preview 05 = Preview