Consider the homogeneous linear third order equation A) xy'''−xy'' + y'−y = 0 Given...
Consider the homogeneous linear third order equation
A) xy'''−xy'' + y'−y = 0
Given that y1(x) = e^x is a solution. Use the substitution y = u*y1 to reduce this third order equation to a homogeneous linear second order equation in the variable w = u'. You do not need to solve this second order equation.
B.) xy''' + (1−x)y'' + xy'−y = 0.
Given that y1(x) = x is a solution. Use the substitution y = uy1 to reduce this third order equation to a homogeneous linear second order equation in the variable w = u0. You do not need to solve this second order equation.
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Consider the homogeneous linear third order equation
A) xy'''−xy'' + y'−y = 0
Given...
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular...
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular solution satisfying the given initial conditions. yl) + 2y'' – y' - 2y = 0; y(0) = 2, y'(0) = 12, y''(0) = 0; Y1 = ex, y2 = e -X, y3 = e - 2x The particular solution is y(x) = .
Consider the nonhomogeneous second order linear equation of the form y" + 2y' + y =...
Consider the nonhomogeneous second order linear equation of the form y" + 2y' + y = g(t). Given that the fundamental solution set of its homogeneous equation is {e**, te' } For each of the parts below, determine the form of particular solution y, that you would use to solve the given equation using the Method of Undetermined Coefficients. DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS. a) y" + 2y' + y = 2te b) y" + 2y' + y...
2. a) (7 pnts) Solve the second order homogeneous linear differential equation y" - y =...
2. a) (7 pnts) Solve the second order homogeneous linear differential equation y" - y = 0. b) (6 pnts) Without any solving, explain how would you change the above differential equation so that the general solution to the homogeneous equation will become c cos x + C sinx. c) (7 pnts) Solve the second order linear differential equation y" - y = 3e2x by using Variation of Parameters. 5. a) (7 pnts) Determine the general solution to the system...
just focus on A,B,D 1. Homogeneous ODE Find a general solution of the linear non-constant coefficient,...
just focus on A,B,D
1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
A third-order homogeneous linear equation and three linearly independent solutions are given below. Find a particular...
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) = .
Given a second order linear homogeneous differential equation a2(x)” + a (x2y + a)(x2y = 0...
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...
3.1: Characteristic polynomial, linear independence 8. Consider the following differential equation: xy" - (x + 1)y...
3.1: Characteristic polynomial, linear independence 8. Consider the following differential equation: xy" - (x + 1)y + y = 0 The functions, Y1 = +1 72 = 4x + 4 are solutions to this differential equation. Find the general solution, or explain why we don't have enough information to do so.
1. The function: y, = e' is a solution of the homogeneous linear equation: y"-2y'+ y...
1. The function: y, = e' is a solution of the homogeneous linear equation: y"-2y'+ y = 0. Use Reduction of Order to find a second linearly independent solution, then write the general solution for the differential equation: y" - 2y'+y=0
1. Consider the equation xy" - 2y' + (2 - x)y = 0,x > 0. We...
1. Consider the equation xy" - 2y' + (2 - x)y = 0,x > 0. We can easily verify that y(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
5) Consider the second order linear non-homogeneous differential equation tay" - 2y = 3t2 - 1,t>...
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)
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Consider the nonhomogeneous second order linear equation of the form y" + 2y' + y = g(t). Given that the fundamental solution set of its homogeneous equation is {e**, te' } For each of the parts below, determine the form of particular solution y, that you would use to solve the given equation using the Method of Undetermined Coefficients. DO NOT ATTEMPT TO SOLVE THE COEFFICIENTS. a) y" + 2y' + y = 2te b) y" + 2y' + y...
2. a) (7 pnts) Solve the second order homogeneous linear differential equation y" - y = 0. b) (6 pnts) Without any solving, explain how would you change the above differential equation so that the general solution to the homogeneous equation will become c cos x + C sinx. c) (7 pnts) Solve the second order linear differential equation y" - y = 3e2x by using Variation of Parameters. 5. a) (7 pnts) Determine the general solution to the system...
just focus on A,B,D
1. Homogeneous ODE Find a general solution of the linear non-constant coefficient, homogeneous ODE for y(x) x3y'" – 3xy" + (6 – x2)xy' – (6 – x?)y = 0 as follows. a) You are given that yı(x) = x is a solution to the above homogeneous ODE. Confirm (by substitution) that this is the case. b) Apply reduction of order to find the remaining two solutions, then state the general solution. (Hint: The substitution y2(x) =...
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) = .
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...
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1. The function: y, = e' is a solution of the homogeneous linear equation: y"-2y'+ y = 0. Use Reduction of Order to find a second linearly independent solution, then write the general solution for the differential equation: y" - 2y'+y=0
1. Consider the equation xy" - 2y' + (2 - x)y = 0,x > 0. We can easily verify that y(x) = e* is a solution of the equation. Use reduction of order to determine the general solution of the equation.
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