Questions and Answers of A First Course in Differential Equations

In problem proceed as in Example 1 to find a particular solution yp(x) of the given differential equation in the integral form (10).y'' - 16y = f(x)
Find the general solution of x4y'' + x3y' - 4x2y = 1 given that y1 = x2 is a solution of the associated homogeneous equation.
Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas.y'' - 2y' + y = 4x2 - 3 + x-1ex
Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas.3y'' - 6y' + 30y = 15 sin x + ex tan 3x
Solve the given third-order differential equation by variation of parameters.y''' - 3y'' + 2y' = e2x/1 + ex
Solve the given third-order differential equation by variation of parameters.y''' – 2y'' – y' + 2y = e4x
Solve the given third-order differential equation by variation of parameters.y''' + 4y' = sec 2x
Solve the given third-order differential equation by variation of parameters.y''' + y' = tan x
In problem the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ∞). Find the general solution of the given nonhomogeneous
In problem the indicated functions are known linearly independent solutions of the associated homogeneous differential equation on (0, ∞). Find the general solution of the given nonhomogeneous
Solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0.y'' - 4y' + 4y = (12x2 - 6x)e2x
Solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0.y'' + 2y' - 8y = 2e-2x – e-x
Solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0.2y'' + y' - y = x + 1
Solve each differential equation by variation of parameters, subject to the initial conditions y(0) = 1, y'(0) = 0.4y'' - y = xex/2
Solve each differential equation by variation of parameters.4y'' – 4y' + y = ex/2√1 – x2
Solve each differential equation by variation of parameters.3y'' - 6y' + 6y = ex sec x
Solve each differential equation by variation of parameters.2y'' + 2y' + y = 4√x
Solve each differential equation by variation of parameters.y'' + 2y' + y = e-t ln t
Solve each differential equation by variation of parameters.y'' - 2y' + y = et arctan t
Solve each differential equation by variation of parameters.y'' + 3y' + 2y = sin ex
Solve each differential equation by variation of parameters.y'' - 2y' + y = ex/1 + x2
Suppose L is a linear differential operator that factors but has variable coefficients. Do the factors of L commute? Defend your answer.
Solve the given initial-value problem.y(4) – y''' = x + ex, y(0) = 0, y'(0) = 0, y''(0) = 0, y'''(0) = 0
Solve the given initial-value problem.y'' - 4y' + 8y = x3, y(0) = 2, y'(0) = 4
Solve the given initial-value problem.y''' - 2y + y' = xex + 5, y(0) = 2, y'(0) = 2, y''(0) = 1
Solve the given initial-value problem.y'' + y = 8 cos 2x -4 sin x,
Solve the given initial-value problem.y'' + 5y' - 6y = 10e2x, y(0) = 1, y'(0) = 1
Solve the given initial-value problem.y'' - 5y' = x - 2, y(0) = 0, y'(0) = 2
Solve the given initial-value problem.y'' + y' = x, y(0) = 1, y'(0) = 0
Solve the given initial-value problem.y'' - 64y = 16, y(0) = 1, y'(0) = 0
Solve the given differential equation by undetermined coefficients.y'' + 25y = 6 sin x
Solve the given differential equation by undetermined coefficients.y(4) - 4y'' = 5x2 - e2x
Solve the given differential equation by undetermined coefficients.y(4) - 2y'' + y = ex + 1
Solve the given differential equation by undetermined coefficients.2y''' - 3y'' - 3y' = 2y = (ex + e-x)2
Solve the given differential equation by undetermined coefficients.y''' - 3y'' + 3y'' - y = ex - x + 16
Solve the given differential equation by undetermined coefficients.y''' – y'' + y' - y = xex – e-x + 7
Solve the given differential equation by undetermined coefficients.y''' + 8y = -6x2 + 9x + 2
Solve the given differential equation by undetermined coefficients.y'' + 4y = cos2x
Solve the given differential equation by undetermined coefficients.y'' + y' + y = x sin x
Solve the given differential equation by undetermined coefficients.y'' + y = 4 cos x - sin x
Solve the given differential equation by undetermined coefficients.y'' + 25y = 20 sin 5x
Solve the given differential equation by undetermined coefficients.y'' + y' + ¼ y = ex(sin 3x - cos 3x)
Solve the given differential equation by undetermined coefficients.y'' - 2y' + 5y = ex sin x
Solve the given differential equation by undetermined coefficients.y'' + 2y' + y = x2e-x
Solve the given differential equation by undetermined coefficients.y'' - y = x2ex + 5
Solve the given differential equation by undetermined coefficients.y'' + 3y' - 10y = x(ex + 1)
Solve the given differential equation by undetermined coefficients.y'' + 6y' + 9y = -xe4x
Solve the given differential equation by undetermined coefficients.y'' + 4y = 4 cos x + 3 sin x - 8
Solve the given differential equation by undetermined coefficients.y'' + 6y' + 8y = 3e-2x + 2x
Solve the given differential equation by undetermined coefficients.y'' + 2y' - 3y = 4ex - 9
Solve the given differential equation by undetermined coefficients.y'' + 2y' + 2y = 5e6x
Solve the given differential equation by undetermined coefficients.y'' – y' - 12y = e4x
Solve the given differential equation by undetermined coefficients.y'' - 2y' + y = x3 + 4x
Solve the given differential equation by undetermined coefficients.y''' + y'' = 8x2
Solve the given differential equation by undetermined coefficients.y'' + 3y' = 4x - 5
Solve the given differential equation by undetermined coefficients.y'' + 4y' + 4y = 2x + 6
Solve the given differential equation by undetermined coefficients.y''' + 2y'' + y' = 10
Solve the given differential equation by undetermined coefficients.y'' + y' = 3
Solve the given differential equation by undetermined coefficients.2y'' - 7y' + 5y = 29
Solve the given differential equation by undetermined coefficients.y'' - 9y = 54
Find linearly independent functions that are annihilated by the given differential operator.D2(D - 5)(D - 7)
Find linearly independent functions that are annihilated by the given differential operator.D3 - 10D2 + 25D
Find linearly independent functions that are annihilated by the given differential operator.D2 - 6D + 10
Find linearly independent functions that are annihilated by the given differential operator.D2 + 5
Find linearly independent functions that are annihilated by the given differential operator.D2 - 9D - 36
Find linearly independent functions that are annihilated by the given differential operator.(D - 6)(2D + 3)
Find linearly independent functions that are annihilated by the given differential operator.D2 + 4D
Find linearly independent functions that are annihilated by the given differential operator.D5
Find a linear differential operator that annihilates the given function.e-x sin x - e2x cos x
Find a linear differential operator that annihilates the given function.3 + ex cos 2x
Find a linear differential operator that annihilates the given function.(2 - ex)2
Find a linear differential operator that annihilates the given function.e-x + 2xex - x2ex
Find a linear differential operator that annihilates the given function.8x - sin x + 10 cos 5x
Find a linear differential operator that annihilates the given function.13x + 9x2 - sin 4x
Find a linear differential operator that annihilates the given function.1 + sin x
Find a linear differential operator that annihilates the given function.cos 2x
Find a linear differential operator that annihilates the given function.x + 3xe6x
Find a linear differential operator that annihilates the given function.1 + 7e2x
Find a linear differential operator that annihilates the given function.x3(1 - 5x)
Find a linear differential operator that annihilates the given function.1 + 6x -2x3
Verify that the given differential operator annihilates the indicated functions.D2 + 64; y = 2 cos 8x – 5 sin 8x
Verify that the given differential operator annihilates the indicated functions.(D – 2)(D – 5); y = e2x + 3e-5x
Verify that the given differential operator annihilates the indicated functions.2D – 1; y = 4ex/2
Verify that the given differential operator annihilates the indicated functions.D4; y = 10x3 – 2x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y(4) + 8y'' + 16y = (x3 – 2x)e4x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y(4) + 8y' = 4
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y''' + 4y'' + 3y' = x2 cos x – 3x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y''' + 2y'' - 13y' + 10y = xe-x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y''' + 4y' = ex cos 2x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y''' + 10y'' + 25y' = ex
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.2y'' – 3y' – 2y = 1
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y'' – 4y' – 12y = x - 6
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.y'' – 5y = x2 – 2x
Write the given differential equation in the form L(y) = g(x), where L is a linear differential operator with constant coefficients. If possible, factor L.9y'' – 4y = sin x
Find a particular solution of the given differential equation. Use a CAS as an aid in carrying out differentiations, simplifications, and algebra.y'' – 4y' + 8y = (2x2 – 3x)e2x cos2x + (10x2 –
Without solving, match a solution curve of y'' + y = f (x) shown in the figure with one of the following functions:(i) f (x) = 1, (ii) f (x) = e-x,(iii) f (x) = ex, (iv) f (x) = sin 2x,(v) f (x) = ex
Discuss how the method of this section can be used to find a particular solution of y'' + y = sin x cos 2x. Carry out your idea.
Consider the differential equation ay'' + by' + cy = ekx, where a, b, c, and k are constants. The auxiliary equation of the associated homogeneous equation isam2 + bm + c = 0.(a) If k is not a root
Solve the given initial-value problem in which the input function g(x) is discontinuous. Solve each problem on two intervals, and then find a solution so that y and y' are continuous at x = π/2
Solve the given initial-value problem in which the input function g(x) is discontinuous. Solve each problem on two intervals, and then find a solution so that y and y' are continuous at x = π/2

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