1. The function: y, = e' is a solution of the homogeneous linear equation: y"-2y'+ y...
5) Consider the second order linear non-homogeneous differential equation tay" - 2y = 3t2 - 1,t> 0. a) Verify that y(t) = t- and y(t) = t-1 satisfy the associated homogeneous equation tay" - 2y = 0. (5 points) b) Find a particular solution to the non-homogeneous differential equation. (10 points) c) Find the general solution to the non-homogeneous differential equation. (5 points)
3. Consider the differential equation ty" - (t+1)y + y = t?e?', t>0. (a) Find a value ofr for which y = et is a solution to the corresponding homogeneous differential equation. (b) Use Reduction of Order to find a second, linearly independent, solution to the correspond- ing homogeneous differential equation. (c) Use Variation of Parameters to find a particular solution to the nonhomogeneous differ- ential equation and then give the general solution to the differential equation.
Problem 1 (14 points) (a) Find the general solution to a third-order linear homogeneous differential equation for y(1) with real numbers as coefficients if two linearly independent solutions are known to be e-21 and sin(3.c). e (b) Determine that differential equation described in part (a).
Problem 1 (14 points) (a) Find the general solution to a third-order linear homogeneous differential equation for y(1) with real numbers as coefficients if two linearly independent solutions are known to be e-21 and sin(3.c). e (b) Determine that differential equation described in part (a).
One of the solutions to the following differential equation (1 – 2x – 2y + 2(1+x)y – 2y = 0 is known to be yı (x) = 1 +1 Find the second linearly independent solution y2 (2) using the method of Reduction of Order.
Given a second order linear homogeneous differential equation а2(х)у" + а (х)У + аo(х)у — 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, V2. But there are times when only one function, call it y, 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 a2(x) F 0 we rewrite...
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...
2) (25 points) a) (5 points) Verify that y= eat is a solution of the homogeneous differential equation y" - 12y' + 36 y = 0. b) (15 points) Use the method of reduction of order to find a second solution 72 of the given homogeneous equation and a particular solution y of the nonhomogeneous differential equation y" - 12y' + 36 y = 36. e) (5 points) Can you write the general solution of the nonhomogeneous differential equation y"...
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions. y (3) + 2y" - y' - 2y = 0; y(0) = 7, y' (0) = 16, y''O) = 0; e y2 = e-X, y3 = e - 2x Y The particular solution is y(x) = .
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