As in the text discussion, suppose that an airplane

Problem 1CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 3CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 1P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x + y) y? = x ? y

Problem 2CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

In the calculus of plane curves, one learns that the curvature k of the curve y = y(x) at the point (x, y) is given by and that the curvature of a circle of radius r is k = 1/r. Conversely substitutes k = 1/r above to derive a general solution of the second-order differential equation (with r constant) in the form . Thus a circle of radius r (or a part thereof) is the only plane curve with constant curvature 1/r.

Problem 2P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. 2xyy? = x2 + 2y2

Problem 3P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout.

Problem 4CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 5CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 6CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 5P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x(x + y)y? = y(x ? y)

Problem 4P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x ? y)y? = x + y

Problem 6P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x + 2y) y? = y

Problem 7CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 7P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. xy2y? = x3 + y3

Problem 8CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 9CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 8P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x2 y? = xy + x2ey/x

Problem 10CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x. = 1 + x2 + y2 + x2y2

Problem 9P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x2y? = xy + y2

Problem 10P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. xyy? = x2 + 3y2

Problem 11CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x. x2y? = xy + 3y2

Problem 12P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout.

Problem 11P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x2 ? y2)y? = 2xy

Problem 12CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 13CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 13P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout.

Problem 14CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 14P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout.

Problem 15CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 15P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x(x + y) y? + y(3x + y) = 0

Problem 16P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout.

Problem 16CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 17CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 18CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 17P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. y? = (4x + y)2

Problem 18P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x + y) y? = 1

Problem 19CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 19P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x2 y? + 2xy = 5y3

Problem 20CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 21CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 20P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. y2y? + 2xy3 = 6x

Problem 21P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. y? = y + y3

Problem 22CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 22P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. x2 y? + 2xy = 5y4

Problem 23CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 23P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. xy? + 6y = 3xy4/3

Problem 24CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 24P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. 2xy? + y3e?2x = 2xy

Problem 25CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 25P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. y2(xy? + y)(1 + x4)1/2 = x

Problem 26CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 26P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. 3y2y? + y3 = e?x

Problem 27CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 27P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. 3xy2 y? = 3x4 + y3

Problem 29P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (2x sin y cosy) y? = 4x2 + sin2y

Problem 29CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 28P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. xey y? = 2(ey + x3e2x)

Problem 28CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 30P Find general solutions of the differential equations in Problems. Primes denote derivatives with respect to x throughout. (x + ey)y? = xe?y ? 1

Problem 30CRP Find general solutions of the differential equations in Problems. Primes denote derivations with respect to x.

Problem 31P In Problems, verify that the given differential equation is exact; then solve it. (2x + 3y) dx + (3x + 2y) dy = 0

Problem 32CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 31CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 32P In Problems, verify that the given differential equation is exact; then solve it. (4x ? y) dx + (6y ? x) dy = 0

Problem 33CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 33P In Problems, verify that the given differential equation is exact; then solve it. (3x2 + 2y2) dx + (4xy + 6y2) dy = 0

Problem 34CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 34P In Problems, verify that the given differential equation is exact; then solve it. (2xy2 + 3x2) dx + (2x2y + 4y3) dy = 0

Problem 35P Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 36CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 35CRP Each of the differential equations in Problems is of two different types considered in this chapter—separable, linear, homogeneous, Bernoulli, exact, etc. Hence, derive general solutions for each of these equations in two different ways; then reconcile your results.

Problem 36P In Problems, verify that the given differential equation is exact; then solve it. (1 + yexy) dx + (2y + xexy) dy = 0

Problem 37P In Problems, verify that the given differential equation is exact; then solve it.

Problem 38P In Problems, verify that the given differential equation is exact; then solve it.

Problem 39P In Problems, verify that the given differential equation is exact; then solve it. (3x2y3 + y4) dx + (3x3y2 + y4 + 4xy3) dy = 0

Problem 41P In Problems, verify that the given differential equation is exact; then solve it.

Problem 40P In Problems, verify that the given differential equation is exact; then solve it. (ex sin y + tan y) dx + (ex cos y + x sec2 y) dy = 0

Problem 42P In Problems, verify that the given differential equation is exact; then solve it.

Problem 44P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. yy? + (y?)2 = 0

Problem 43P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. xy? = y?

Problem 45P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. y? + 4y = 0

Problem 46P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. xy? + y? = 4x

Problem 47P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. y? = (y?)2

Problem 48P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. x2y? + 3xy? = 2

Problem 49P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. yy? + (y?)2 = yy?

Problem 51P y? = 2y(y?)3

Problem 50P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. y? = (x + y’)2

Problem 52P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. y3 y? = 1

Problem 55P Show that the substitution v = ax + by + c transforms the differential equation dy/dx = F(ax + by + c) into a separable equation.

Problem 53P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. y? = 2yy?

Problem 54P Find a general solution of each reducible second-order differential equation in Problems. Assume x, y and/or y? positive where helpful. yy? = 3(y?)2

Problem 56P Suppose that n ? 0 and n?1. Show that the substitution v = y1-n transforms the Bernoulli equation into the linear equation.

Problem 57P Show that the substitution v = ln y transforms the differential equation into the linear equation

Problem 58P Use the idea in Problem to solve the equation Problem Show that the substitution v = ln y transforms the differential equation into the linear equation

Problem 59P Solve the differential equation by finding h and k so that the substitutions x = u + h, y = v + k transform it into the homogeneous equation

Problem 62P Show that the solution curves of the differential equation are of the form x3 + y3 = 3Cxy.

Problem 60P Use the method in Problem to solve the differential equation

Problem 63P The equation is called a Riccati equation. Suppose that one particular solution y1(x) of this equation is known. Show that the substitution transforms the Riccati equation into the linear equation

Problem 61P Make an appropriate substitution to find a solution of the equation Does this general solution contain the linear solution that is readily verified by substitution in the differential equation?

Problem 64P Use the method of Problem to solve the equations in Problems 64 atid 65, given that y1(c) = x is a solution of each.

Problem 65P Use the method of Problem to solve the equations in Problems 64 atid 65, given that y1(c) = x is a solution of each.

Problem 66P An equation of the form is called a Clairaut equation. Show that the one-parameter family of straight lines described by is a general solution of Eq.

Problem 68P Derive Eq. (18) in this section from Eqs. (16) and (17)

Problem 69P In the solution of Example, suppose thata = 100 mi, v0 = 400 mi/h, and w = 40 mi/h. Now how far northward does the wind blow the airplane?

Problem 70P As in the text discussion, suppose that an airplane maintains a heading toward an airport at the origin. If v0 = 500 mi/h and w = 50 mi/h (with the wind blowing due north), and the plane begins at the point (200, 150), show that its trajectory is described by

Problem 67P Consider the Clairaut equation for which in Eq. Show that the line is tangent to the parabola y = x2 at the point Explain why this implies that y = x2 is a singular solution of the given Clairaut equation. This singular solution and the one-parameter family of straight line solutions are illustrated in Fig. 1.6.10

Problem 71P A river 100 ft wide is Mowing north atw feet per second. A dog starts at(100. 0) and swims at u0= 4 ft/s, always heading toward a tree at (0, 0) on the west bank directly across from the dog’s starting point, (a) If w = 2 ft/s, show that the dog reaches the tree, (b) If w = 4 ft/s? show that the dog reaches instead the point on the west bank 50 ft north of the tree, (c) If w = 6 ft/s, show that the doe never reaches the west bank.