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mathematics
first course differential equations

Questions and Answers of First Course Differential Equations

In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your
Arthur Clarke’s The Wind from the Sun (1963) describes Diana, a spacecraft propelled by the solar wind. Its aluminized sail provides it with a constant acceleration of 0.001g = 0.0098 m/s2. Suppose
In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your
A bomb is dropped from a helicopter hovering at an altitude of 800 feet above the ground. From the ground directly beneath the helicopter, a projectile is fired straight upward toward the bomb,
In Problems 37 through 42, determine by inspection at least one solution of the given differential equation. That is, use your knowledge of derivatives to make an intelligent guess. Then test your
In problem solve the given initial-value problem.y'' + y' = x, y(0) = 1, y'(0) = 0
In problem solve the given initial-value problem.y'' + y = 8 cos 2x ‑ 4 sin x, y(π/2) = -1, y'(π/2) = 0
In problem solve the given initial-value problem.y'' ‑ 5y' = x ‑ 2, y(0) = 0, y'(0) = 2
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 cos x + c2 sin x + c3 cos 2x + c4 sin 2x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 + c2x + c3e8x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 + c2e2x cos 5x + c3e2x sin 5x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1e-x cos x + c2e-x sin x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 cosh 7x + c2 sinh 7x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 cos 3x + c2 sin 3x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1e10x + c2xe10x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1 + c2e2x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1e-4x + c2e-3x
In problem find a homogeneous linear differential equation with constant coefficients whose general solution is given.y = c1ex + c2e5x
In problem figure represents the graph of a particular solution of one of the following differential equations:(a) y'' - 3y' - 4y = 0(b) y'' + 4y = 0(c) y'' + 2y' + y = 0(d) y''
In problem figure represents the graph of a particular solution of one of the following differential equations:(a) y'' - 3y' - 4y = 0(b) y'' + 4y = 0(c) y'' + 2y' + y = 0(d) y''
In problem figure represents the graph of a particular solution of one of the following differential equations:(a) y'' - 3y' - 4y = 0(b) y'' + 4y = 0(c) y'' + 2y' + y = 0(d) y''
In problem solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11).y'' - y = 0, y(0) = 1, y'(1) = 0
In problem solve the given problem first using the form of the general solution given in (10). Solve again, this time using the form given in (11).y'' - 3y = 0, y(0) = 1, y'(0) = 5
In problem solve the given boundary-value problem.y'' - 2y' + 2y = 0, y(0) = 1, y(π) = 1
In problem solve the given boundary-value problem.y'' + y = 0, y'(0) = 0, y'(π/2) = 0
In problem solve the given boundary-value problem.y'' + 4y = 0, y(0) = 0, y(π) = 0
In problem solve the given boundary-value problem.y'' - 10y' + 25y = 0, y(0) = 1, y(1) = 0
In problem solve the given initial-value problem.y''' + 2y'' - 5y' - 6y = 0, y(0) = y'(0) = 0, y''(0) = 1
In problem solve the given initial-value problem.y''' + 12y'' + 36y' = 0, y(0) = 0, y'(0) = 1, y''(0) = -7
In problem solve the given initial-value problem.y'' - 2y' + y = 0, y(0) = 5, y'(0) = 10
In problem solve the given initial-value problem.y'' + y' + 2y = 0, y(0) = y'(0) = 0
In problem solve the given initial-value problem.4y'' ‑ 4y' ‑ 3y = 0, y(0) = 1, y'(0) = 5
In problem solve the given initial-value problem.d2y/dt2 – 4 dy/dt - 5y = 0, y(1) = 0, y'(1) = 2
In problem solve the given initial-value problem.d2y/dθ2 + y = 0, y(π/3) = 0, y'(π/3) = 2
In problem solve the given initial-value problem.y'' + 16y = 0, y(0) = 2, y'(0) = ‑2
In problem find the general solution of the given higher order differential equation.2 d5x/ds5 – 7 d4x/ds4 + 12 d3x/ds3 + 8 d2x/ds2 = 0
In problem find the general solution of the given higher order differential equation.d5u/dr5 + 5 d4u/dr4 – 2 d3u/dr3 – 10 d2u/dr2 + du/dr + 5u = 0
In problem find the general solution of the given higher order differential equation.d4y/dx4 – 7 d2y/dx2 - 18y = 0
In problem find the general solution of the given higher order differential equation.16 d4y/dx4 + 24 d2y/dx2 + 9y = 0
In problem find the general solution of the given higher order differential equation.y(4) ‑ 2y'' + y = 0
In problem find the general solution of the given higher order differential equation.y(4) + y''' + y'' = 0
In problem find the general solution of the given higher order differential equation.y''' ‑ 6y'' + 12y' ‑ 8y = 0
In problem find the general solution of the given higher order differential equation.y''' + 3y'' + 3y' + y = 0
In problem find the general solution of the given higher order differential equation.d3x/dt3 - d2x/dt2 - 4x = 0
In problem find the general solution of the given higher order differential equation.d3u/dt3 + d2u/dt2 - 2u = 0
In problem find the general solution of the given higher order differential equation.y''' + 3y'' ‑ 4y' ‑ 12y = 0
In problem find the general solution of the given higher order differential equation.y''' ‑ 5y'' + 3y' + 9y = 0
In problem find the general solution of the given higher order differential equation.y''' ‑ y = 0
In problem find the general solution of the given higher order differential equation.y''' ‑ 4y'' ‑ 5y' = 0
In problem find the general solution of the given second-order differential equation.2y'' ‑ 3y' + 4y = 0
In problem find the general solution of the given second-order differential equation.3y'' + 2y' + y = 0
In problem find the general solution of the given second-order differential equation.2y'' + 2y' + y = 0
In problem find the general solution of the given second-order differential equation.y'' ‑ 4y' + 5y = 0
In problem find the general solution of the given second-order differential equation.3y'' + y = 0
In problem find the general solution of the given second-order differential equation.y'' + 9y = 0
In problem find the general solution of the given second-order differential equation.y'' + 4y' ‑ y = 0
In problem find the general solution of the given second-order differential equation.12y'' ‑ 5y' ‑ 2y = 0
In problem find the general solution of the given second-order differential equation.y'' + 8y' + 16y = 0
In problem find the general solution of the given second-order differential equation.y'' – y' ‑ 6y = 0
In problem find the general solution of the given second-order differential equation.y'' ‑ 36y = 0
In problem find the general solution of the given second-order differential equation.4y'' + y' = 0
(a) Verify that y1(x) = exis a solution ofxy'' €‘ (x + 10)y' + 10y = 0.(b) Use (5) to find a second solution y2(x). Use a CAS to carry out the required integration.(c)
Verify that y1(x) = x is a solution of xy'' – xy' + y = 0. Use reduction of order to find a second solution y2(x) in the form of an infinite series. Conjecture an interval of definition
(a) Give a convincing demonstration that the second-order equation ay'' + by' + cy = 0, a, b, and c constants, always possesses at least one solution of the form y1 = em1x, m1 a constant.(b)
In problem the indicated function y1(x) is a solution of the given differential equation. Use formula (5) to find a second solution y2(x) expressed in terms of an integral defined function. See
In problem the indicated function y1(x) is a solution of the given differential equation. Use formula (5) to find a second solution y2(x) expressed in terms of an integral defined function. See
In problem the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of‑ reduction of order to find a second solution y2(x) of the homogeneous equation and
In problem the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of‑ reduction of order to find a second solution y2(x) of the homogeneous equation and
In problem the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of‑ reduction of order to find a second solution y2(x) of the homogeneous equation and
In problem the indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of‑ reduction of order to find a second solution y2(x) of the homogeneous equation and
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).(1 ‑ x2)y'' + 2xy'
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).(1 ‑ 2x ‑ x2)y''
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).x2y'' ‑ 3xy' + 5y
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).x2y'' – xy' + 2y
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).4x2y'' + y = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).xy'' + y' = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).x2y'' + 2xy' ‑ 6y
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).x2y'' ‑ 7xy' + 16y
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).6y'' + y' ‑ y =
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).9y'' ‑ 12y' + 4y
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' ‑ 25y = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' ‑ y = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' + 9y = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' + 16y = 0; y1
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' + 2y' + y = 0;
In problem the indicated function y1(x) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution y2(x).y'' - 4y' + 4y = 0;
Suppose that y1, y2, . . . , yk are k nontrivial solutions of a homogeneous linear nth-order differential equation with constant coefficients and that k = n + 1. Is the set of solutions y1, y2, .
Suppose y1, y2, . . . , yk are k linearly independent solutions on (‑∞, ∞) of a homogeneous linear nth-order differential equation with constant coefficients. By Theorem 4.1.2 it follows that
Is the set of functions f1(x) = ex+2, f2(x) = ex‑3 linearly dependent or linearly independent on (‑∞, ∞)? Discuss.
(a) Verify that y1 = x3 and y2 = |x|3 are linearly independent solutions of the differential equation x2y'' ‑ 4xy' + 6y = 0 on the interval (‑∞, ∞).(b) For the functions y1 and y2 in
Suppose that y1 = ex and y2 = e‑x are two solutions of a homogeneous linear differential equation. Explain why y3 = cosh x and y4 sinh x are also solutions of the equation.
Let n 1, 2, 3, . . . . Discuss how the observations Dnxn‑1 = 0 and Dnxn = n! can be used to find the general solutions of the given differential equations.(a) y'' = 0(b) y''' = 0(c)
(a) By inspection nd a particular solution of y' + 2y = 10.(b) By inspection nd a particular solution of y'' + 2y = ‑4x.(c) Find a particular solution of y'' + 2y = ‑4x + 10. (d)
(a) Verify that yp1 = 3e2x and yp2 = x2 + 3x are, respectively, particular solutions of y'' - 6y' + 5y = -9e2x and y'' - 6y' + 5y = 5x2 + 3x - 16.(b) Use part (a) to find particular solutions of
In problem verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.2x2y'' + 5xy' + y = x2 ‑ x;y =
In problem verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.y'' ‑ 4y' + 4y = 2e2x + 4x ‑
In problem verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.y'' + y = sec x;y = c1 cos x + c2
In problem verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.y'' ‑ 7y' + 10y = 24ex;y = c1e2x
In problem verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.y(4) + y'' = 0; 1, x, cos x, sin x,
In problem verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.x3y''' + 6x2y'' + 4xy' ‑ 4y = 0;

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