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study help
mathematics
first course differential equations
Questions and Answers of
First Course Differential Equations
Falling Chain A portion of a uniform chain of length 8ft is loosely coiled around a peg at the edge of a high horizontal platform, and the remaining portion of the chain hangs at rest
True or False: Every separable first-order equation dy/dx = g(x)h(y) is exact.
Differential equations are sometimes solved by having a clever idea. Here is a little exercise in cleverness: Although the differential equation {x – √(x2 + y2)} dx + y dy = 0 is not exact, show
Discuss how the functions M(x, y) and N(x, y) can be found so that each differential equation is exact. Carry out your ideas.(a) M(x, y) dx + (xexy + 2xy + 1/x) dy = 0(b) {x-1/2y1/2 + x/(x2 + y)} dx
Discuss why we can conclude that the interval of definition of the explicit solution of the IVP (the blue curve in the following figure) is (21, 1). y
Consider the concept of an integrating factor. Are the two equations M dx + N dy = 0 and μM dx + μN dy = 0 necessarily equivalent in the sense that a solution of one is also a solution of the
(a) Show that a one-parameter family of solutions of the equation(4xy + 3x2) dx + (2y + 2x2) dy = 0 is x3 + 2x2y + y2 = c.(b) Show that the initial conditions y(0) = -2 and y(1) = 1 determine the
Solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor.(x2 + y2 - 5) dx = (y + xy) dy, y(0) = 1
Solve the given initial-value problem by finding, as in Example 4, an appropriate integrating factor.x dx + (x2y + 4y) dy = 0, y(4) = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.(y2 + xy3) dx + (5y2 - xy + y3 sin y) dy = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.(10 - 6y + e-3x) dx - 2 dy = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.cos x dx + (1 + 2/y) sin x dy = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.6xy dx + (4y + 9x2) dy = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.y(x + y + 1) dx + (x + 2y) dy = 0
Solve the given differential equation by finding, as in Example 4, an appropriate integrating factor.(2y2 + 3x) dx + 2xy dy = 0
Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor μ(x, y) and verify that the new equation is exact. Solve.(x2 +
(a) Use a CAS to graph the solution curve of the initial-value problem in Problem 47 on the interval [0, ).(b) Use a CAS to nd the value of the absolute maximum of the solution y(x)
Find the value of k so that the given differential equation is exact.(6xy3 + cos y) dx + (2kx2y2 - x sin y) dy = 0
Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor μ(x, y) and verify that the new equation is exact. Solve.(-xy
Find the value of k so that the given differential equation is exact.(y3 + kxy4 - 2x) dx + (3xy2 + 20x2y3) dy = 0
Solve the given initial-value problem.{(1/1 + y2)} + cos x - 2xy) dy/dx = y(y + sin x), y(0) = 1
Solve the given initial-value problem.(y2 cos x - 3x2y - 2x) dx + (2y sin x - x3 + ln y) dy = 0, y(0) = e
Solve the given initial-value problem.(3y2 - t2/y5) – dy/dt + t/2y4 = 0, y(1) = 1
Solve the given initial-value problem.(4y + 2t - 5) dt + (6y + 4t - 1) dy = 0, y(-1) = 2
Solve the given initial-value problem.(ex + y) dx + (2 + x + yey) dy = 0, y(0) = 1
Solve the given initial-value problem.(x + y)2 dx + (2xy + x2 - 1) dy = 0, y(1) = 1
Determine whether the given differential equation is exact. If it is exact, solve it.{1/t + 1/t2 – y/(t2 + y2)} dt + {yey + t/(t2 + y2)} dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(4t3y - 15t2 - y) dt + (t4 + 3y2 -t) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(2y sin x cos x - y + 2y2exy2) dx = (x - sin2 x - 4xyexy2 ) dy
Determine whether the given differential equation is exact. If it is exact, solve it.(tan x - sin x sin y) dx + cos x cos y dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(5y - 2x)y9 - 2y = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(x2y3 – 1/1 + 9x2) dx/dy + x3y2 = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(1 – 3/y + x) dy/dx + y = 3/x - 1
Determine whether the given differential equation is exact. If it is exact, solve it.x dy/dx = 2xex - y + 6x2
Determine whether the given differential equation is exact. If it is exact, solve it.(3x2y + ey) dx + (x3 + xey - 2y) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(y ln y – e-xy) dx + (1/y + x ln y) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(x3 + y3) dx + 3xy2 dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(x - y3 + y2 sin x) dx = (3xy2 + 2y cos x) dy
Determine whether the given differential equation is exact. If it is exact, solve it.(1 + ln x + y/x)dx = (1 - ln x) dy
Determine whether the given differential equation is exact. If it is exact, solve it.(x2 - y2) dx + (x2 - 2xy) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(2y – 1/x + cos 3x)dy/dx + y/x2 – 4x3 + 3y sin 3x = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(2xy2 - 3) dx + (2x2y + 4) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(sin y - y sin x) dx + (cos x - x cos y - y) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(5x + 4y) dx + (4x - 8y3) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(2x + y) dx - (x + 6y) dy = 0
Determine whether the given differential equation is exact. If it is exact, solve it.(2x - 1) dx + (3y + 7) dy = 0
(a) Use a CAS to graph the solution curve of the initial-value problem in Problem 48 on the interval (-, ).(b) It is known that Fresnel sine integral S(x) 1/2 as
(a) Use a CAS to graph the solution curve of the initial-value problem in Problem 44 on the interval (-∞, ∞).(b) Use tables or a CAS to value the value y(2).Data from problem 44Proceed as in
Heart Pacemaker A heart pacemaker consists of a switch, a battery of constant voltage E0, a capacitor with constant capacitance C, and the heart as a resistor with constant resistance R. When the
The following system of differential equations is encountered in the study of the decay of a special type of radioactive series of elements:dx/dt = λ1xdy/dt = λ1x – λ2y,where λ1 and λ2 are
Suppose P(x) is continuous on some interval I and α is a number in I. What can be said about the solution of the initial-value problem y' + P(x)y = 0, y(a) = 0?
In determining the integrating factor (3), we did not use a constant of integration in the evaluation of ∫P(x) dx. Explain why using ∫P(x) dx + c1 has no effect on the solution of (2).
(a) Construct a linear first-order differential equation of the form xy' + 3y = g(x) for which y = x3 1 c/x3 is its general solution. Give an interval I of definition of this solution.(b) Give an
Use a graphing utility to graph the continuous function y(x).dy/dx + 2xy = f (x), y(0) = 2, where
Use a graphing utility to graph the continuous function y(x).dy/dx + y = f (x), y(0) = 1, where 1, f(x) = -1,
Use a graphing utility to graph the continuous function y(x).dy/dx + 2y = f (x), y(0) = 0, where |1, 0sx
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.y' + (tan x)y = cos2x, y(0) = -1
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.y' - (sin x)y = 2 sin x, y(π/2) = 1
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.x(x + 1) dy/dx + xy = 1, y(e) = 1
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.(x + 1) dy/dx + y = ln x, y(1) = 10
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.y' + 4xy = x3ex2, y(0) = -1
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.x dy/dx + y = 4x + 1, y(1) = 8
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.dT/dt = k(T - Tm), T(0) = T0, k, Tm, T0 constants
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.L di/dt + Ri = E, i(0) = i0, L, R, E, i0 constants
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.y dx/dy - x = 2y2, y(1) = 5
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.y dx/dy - x = 2y2, y(1) = 5
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.xy' + y = ex, y(1) = 2
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.dy/dx = 2x - 3y, y(0) = 1/3
Solve the given initial-value problem. Give the largest interval I over which the solution is defined.dy/dx = x + 5y, y(0) = 3
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
Find the general solution of the given differential equation. Give the largest interval I over which the general solution is defined. Determine whether there are any transient terms in the
(a) Use a CAS and the concept of level curves to plot representative graphs of members of the family of solutions of the differential equationdy/dx = x(1 - x)/y(-2 + y).Experiment with different
(a) Find an implicit solution of the IVP(2y + 2)dy - (4x3 + 6x)dx = 0, y(0) = -3.(b) Use part (a) to nd an explicit solution y = -(x) of the IVP.(c) Consider your answer to part (b) as a function
(a) Use a CAS and the concept of level curves to plot representative graphs of members of the family of solutions of the differential equationdy/dx = - (8x + 5) / (3y – 11). Experiment with
In (16) of Section 1.3 we saw that a mathematical model for the shape of a exible cable strung between two vertical supports isdy/dx = W/T1
(a) The differential equation in Problem 27 is equivalent to the normal formDy/dx = √(1 - y2) / (1 - x2) in the square region in the xy-plane dened by |x| < 1 < |y| , 1. But the
Find a function whose square plus the square of its derivative is 1.
(a) Solve the two initial-value problems:Dy/dx = y, y(0) = 1anddy/dx = y + y/x ln x, y(e) = 1.(b) Show that there are more than 1.65 million digits in the y-coordinate of the point of intersection of
We saw that every autonomous rstorder differential equation dy/dx = f (y) is separable. Does this fact help in the solution of the initial-value problemDy/dx = √(1+ y2) sin2 y, y(0) = ½
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