Questions and Answers of A First Course in Differential Equations

If y = sin 5x is a solution of a homogeneous linear second-order differential with constant coefficients, then the general solution of the DE is _______.Answer problem without referring back to the
If the set consisting of two functions f1 and f2 is linearly independent on an interval I, then the Wronskian W(f1, f2) ≠ 0 for all x in I. __________Answer problem without referring back to the
A constant multiple of a solution of a linear differential equation is also a solution. __________Answer problem without referring back to the text. Fill in the blank or answer true or false.
For the method of undetermined coefficients, the assumed form of the particular solution yp for y'' - y = 1 + ex is __________.Answer problem without referring back to the text. Fill in the blank or
The only solution of the initial-value problem y'' + x2y = 0, y(0) = 0, y'(0) = 0 is __________.Answer problem without referring back to the text. Fill in the blank or answer true or false.
A mathematical model for the position x(t) of a moving object isd2x/dt2 + sin x = 0.Use a numerical solver to graphically investigate the solutions of the equation subject to x(0) = 0, '(0) = x1, x1
A mathematical model for the position x(t) of a body moving rectilinearly on the x-axis in an inverse-square force field is given bySuppose that at t = 0 the body starts from rest from the position x
Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation y'' = 2x(y')2 in Example 1. Suppose that –c21 is used as the constant of integration
Discuss how the method of reduction of order considered in this section can be applied to the third-order differential equation y''' = √1 + (y'')2. Carry out your ideas and solve the equation.
In Problem 1 we saw that cos x and ex were solutions of the nonlinear equation (y'')2 - y2 = 0. Verify that sin x and e-x are also solutions. Without attempting to solve the differential equation,
In calculus the curvature of a curve that is defined by a function y = f (x) is defined as
In problem proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility
In problem proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility
In problem proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility
In problem proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at 0, of the given initial-value problem. Use a numerical solver and a graphing utility
In problem show that the substitution u = y' leads to a Bernoulli equation. Solve this equation (see Section 2.5).xy'' = y' + x(y')2
In problem show that the substitution u = y' leads to a Bernoulli equation. Solve this equation (see Section 2.5).xy'' = y' + (y')3
Find two solutions of the initial-value problem
Solve the given initial-value problem.y'' + x(y')2 = 0, y (1) = 4, y'(1) = 2
Solve the given initial-value problem.2y'y'' = 1, y(0) = 2, y'(0) = 1
In problem solve the given differential equation by using the substitution u = y'.y2y'' = y'
In problem solve the given differential equation by using the substitution u = y'.y'' + 2y(y')3 = 0
In problem solve the given differential equation by using the substitution u = y'.(y + 1)y'' = (y')2
In problem solve the given differential equation by using the substitution u = y'.x2y'' + (y')2 = 0
In problem solve the given differential equation by using the substitution u = y'.y'' = 1 + (y')2
In problem solve the given differential equation by using the substitution u = y'.y'' + (y')2 + 1 = 0
Verify that y1 and y2 are solutions of the given differential equation but that y = c1y1 + c2y2 is, in general, not a solution.yy'' = ½ (y')2; y1 = 1, y2 = x2
Verify that y1 and y2 are solutions of the given differential equation but that y = c1y1 + c2y2 is, in general, not a solution.(y'')2 = y2; y1 = ex, y2 = cos x
(a) Reread Problem 8 of Exercises 3.3. In that problem you were asked to show that the system of differential equationsis a model for the amounts of salt in the connected mixing tanks A, B, and C
Reexamine Figure 4.9.1 in Example 3. Then use a rootfinding application to determine when tank B contains more salt than tank A.Figure 4.9.1 25 20 15 10 F 20 40 60 80 100 time pounds of salt
Examine and discuss the following system:Dx – 2Dy = t2(D + 1)x – 2(D + 1)y = 1
Determine a system of differential equations that describes the path of motion in Problem 23 if air resistance is a retarding force k (of magnitude k) acting tangent to the path of the projectile but
A projectile shot from a gun has weight w = mg and velocity v tangent to its path of motion. Ignoring air resistance and all other forces acting on the projectile except its weight, determine a
In problem solve the given initial-value problem. dx = y - 1 dt dy —Зх + 2у dt x(0) = 0, y(0) = 0
In problem solve the given initial-value problem. dx -5x - y dt dy = 4x – y dt x(1) = 0, y(1) = 1
Solve the given system of differential equations by systematic elimination. dx = -x + 3 dt dy -y + z dt dz -x +y dt
Solve the given system of differential equations by systematic elimination. dx бу dt dy = x + z dt dz = x + y dt
Solve the given system of differential equations by systematic elimination.Dx + z = et(D - 1)x + Dy + Dz = 0x + 2y + Dz = et
Solve the given system of differential equations by systematic elimination.Dx = yDy = zDz = x
Solve the given system of differential equations by systematic elimination.D2x - 2(D2 + D)y - sin tx + Dy = 0
Solve the given system of differential equations by systematic elimination.(D - 1)x + (D2 + 1)y = 1(D2 - 1)x + (D + 1)y = 2
Solve the given system of differential equations by systematic elimination. dx dy = e! dt dt dx dx + x + y = 0 dt dr
Solve the given system of differential equations by systematic elimination. dx dy dt 5x = e dt dx dy = 5e dt dt
Solve the given system of differential equations by systematic elimination.(2D2 - D - 1)x - (2D + 1)y = 1(D - 1)x + Dy = -1
Solve the given system of differential equations by systematic elimination.(D2 - 1)x - y = 0(D - 1)x + Dy = 0
Solve the given system of differential equations by systematic elimination.D2x - Dy = t(D + 3)x + (D + 3)y = 2
Solve the given system of differential equations by systematic elimination.Dx + D2y = e3t(D + 1)x = (D - 1)y = 4e3t
Solve the given system of differential equations by systematic elimination. d?x dy dt dr -5x dx, dy -x + 4y dt dt
Solve the given system of differential equations by systematic elimination. d?x 4y + e dr d'y = 4x – e dr
Solve the given system of differential equations by systematic elimination.(D + 1)x + (D - 1)y = 23x + (D + 2)y = -1
Solve the given system of differential equations by systematic elimination.(D2 + 5)- = 2y = 0-2x + (D2 + 2)y = 0
Solve the given system of differential equations by systematic elimination.dx/dt – 4y = 1dy/dt + x = 2
Solve the given system of differential equations by systematic elimination.dx/dt = -y + tdy/dt = x – t
Solve the given system of differential equations by systematic elimination.dx/dt = 4x + 7ydy/dt = x - 2y
Solve the given system of differential equations by systematic elimination.dx/dt = 2x – ydy/dt = x
Use the result in Problem 45 to solvey'' + y = 1, y(0) = 5, y(1) = -10.
Suppose the solution of the boundary-value problemy'' + Py' + Qy = f(x), y(a) = 0, y(b) = 0,a < b, is given by yp(x) = ∫baG(x,t)f(t)dt where y1(x) and y2(x) are solutions of the associated
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.x2y" - 4xy' + 6y = x4, y(1) - y'(1) = 0, y(3) = 0
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.x2y" + xy' = 1, y(e-1) = 0 , y(1) = 0
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.y" - y' = e2x, y(0) = 0, y(1) = 0
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.y" - 2y' + 2y = ex, y(0) = 0, y(π/2) = 0
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.y'' + 9y = 1, y(0) = 0, y'(π) = 0
In problem proceed as in Examples 7 and 8 to find a solution of the given boundary value problem.y'' + y = 1, y(0) = 0, y(1) = 0
In problem 36 find a solution of the BVP when f(x) = x.Problem 36In problems,(a) Use (27) and (28) to find a solution of the boundary-value problem.(b) Verify that the function yp(x) satisfies the
In Problem 35 find a solution of the BVP when f(x) = 1.Problem 35In problems,(a) Use (27) and (28) to find a solution of the boundary-value problem.(b) Verify that the function yp(x) satisfies the
In problems,(a) Use (27) and (28) to find a solution of the boundary-value problem.(b) Verify that the function yp(x) satisfies the differential equations and both boundary-conditions.y'' = f(x),
In problems,(a) Use (27) and (28) to find a solution of the boundary-value problem.(b) Verify that the function yp(x) satisfies the differential equations and both boundary-conditions.y'' = f(x),
In problem proceed as in Example 6 to find a solution of the initial-value problem with the given piecewisedefined forcing function.
In problem proceed as in Example 6 to find a solution of the initial-value problem with the given piecewisedefined forcing function.
In problem proceed as in Example 6 to find a solution of the initial-value problem with the given piecewisedefined forcing function.
In problem proceed as in Example 6 to find a solution of the initial-value problem with the given piecewisedefined forcing function.
In problem proceed as in Example 5 to find a solution of the given initial-value problem.x2y'' – xy' + y = x2, y(1) = 4, y'(1) = 3
In problem proceed as in Example 5 to find a solution of the given initial-value problem.x2y'' - 6y = ln x, y(1) = 1, y'(1) = 3
In problem proceed as in Example 5 to find a solution of the given initial-value problem.x2y'' - 2xy' + 2y = x ln x, y(1) = 1, y'(1) = 0
In problem proceed as in Example 5 to find a solution of the given initial-value problem.x2y'' - 2xy' + 2y = x, y(1) = 2, y'(1) = -1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' + 3y' + 2y = 1/1 + ex, y(0) = 0, y'(0) = 1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' + 3y' + 2y = sin ex, y(0) = -1, y'(0) = 0
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' + y = sec2x, y(π) = 1/2, y'(π) = -1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' + y = csc x cot x, y(π/2) = -π/2, y'(π/2) = -1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' + 6y' + 9y = x, y(0) = 1, y'(0) = -3
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' - 10y' + 25y = e5x, y(0) = -1, y'(0) = 1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' – y' = 1, y(0) = 10, y'(0) = 1
In problem proceed as in Example 5 to find a solution of the given initial-value problem.y'' - 4y = e2x, y(0) = 1, y'(0) = -4
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' + y = sec2x, y(π) = 0, y'(π) = 0
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' + y = csc x cot x, y(π/2) = 0, y'(π/2) = 0
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' + 6y' + 9y = x, y(0) = 0, y'(0) = 0
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' - 10y' + 25y = e5x, y(0) = 0, y'(0) = 0
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' – y' = 1, y(0) = 0, y'(0) = 0
In problem proceed as in Example 3 to find a solution of the given initial-value problem. Evaluate the integral that defines yp(x).y'' - 4y = e2x, y(0) = 0, y'(0) = 0
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).y'' - 2y' + 2y = cos2x
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).y'' + 9y = x + sin x
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).4y'' - 4y' + y = arctan x
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).y'' + 2y' + y = e-x
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).y'' + 3y' - 10y = x2
In problem proceed as in Example 2 to find the general solution of the given differential equation. Do not evaluate the integral that defines yp(x).y'' - 16y = xe-2x
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'' - 2y' + 2y = f(x)
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'' + 9y = f(x)
In problem proceed as in Example 1 to find a particular solution yp(x) of the given differential equation in the integral form (10).4y'' - 4y' + y = f(x)
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'' + 2y' + y = f(x)
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'' + 3y' - 10y = f(x)

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