Answer: In 3944, is a twoparameter family of solutions of the second-order DE If

In Problems 1 and 2, y  1(1  c1ex ) is a one-parameter family of solutions of the first-order DE y  y  y2 . Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.y(0)  13

In Problems 1 and 2, y  1(1  c1ex ) is a one-parameter family of solutions of the first-order DE y  y  y2 . Find a solution of the first-order IVP consisting of this differential equation and the given initial condition.2. y(1)  2

In Problems 36, y  1(x2  c) is a one-parameter family of solutions of the first-order DE y  2xy2  0. Find a y(0)  1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 3

In Problems 36, y  1(x2  c) is a one-parameter family of solutions of the first-order DE y  2xy2  0. Find a y(0)  1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 4

In Problems 36, y  1(x2  c) is a one-parameter family of solutions of the first-order DE y  2xy2  0. Find a y(0)  1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 5

In Problems 36, y  1(x2  c) is a one-parameter family of solutions of the first-order DE y  2xy2  0. Find a y(0)  1 3 solution of the first-order IVP consisting of this differential equation and the given initial condition. Give the largest interval I over which the solution is defined 6

In Problems 710, x  c1 cos t  c2 sin t is a two-parameter family of solutions of the second-order DE x  x  0. Find y(1 2)  4 y(2)  1 2 y(2)  1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.

In Problems 710, x  c1 cos t  c2 sin t is a two-parameter family of solutions of the second-order DE x  x  0. Find y(1 2)  4 y(2)  1 2 y(2)  1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.

In Problems 710, x  c1 cos t  c2 sin t is a two-parameter family of solutions of the second-order DE x  x  0. Find y(1 2)  4 y(2)  1 2 y(2)  1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.

In Problems 710, x  c1 cos t  c2 sin t is a two-parameter family of solutions of the second-order DE x  x  0. Find y(1 2)  4 y(2)  1 2 y(2)  1 3 Copyright 2012 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial rea solution of the second-order IVP consisting of this differentialequation and the given initial conditions.

In Problems 1114, y  c1ex  c2ex is a two-parameter family of solutions of the second-order DE y  y  0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

In Problems 1114, y  c1ex  c2ex is a two-parameter family of solutions of the second-order DE y  y  0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

In Problems 1114, y  c1ex  c2ex is a two-parameter family of solutions of the second-order DE y  y  0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

In Problems 1114, y  c1ex  c2ex is a two-parameter family of solutions of the second-order DE y  y  0. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.

In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IV .

In Problems 15 and 16 determine by inspection at least two solutions of the given first-order IV .

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 1724 determine a region of the xy-plane for which the given differential equation would have a unique solution whose graph passes through a point (x0, y0) in the region.

In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.(1, 4)

In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.(5, 3)

In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.. (2, 3)

In Problems 2528 determine whether Theorem 1.2.1 guarantees that the differential equation possesses a unique solution through the given point.. (1, 1)

(a) By inspection find a one-parameter family of solutions of the differential equation xy  y. Verify that each member of the family is a solution of the initial-value problem xy  y, y(0)  0. (b) Explain part (a) by determining a region R in the xy-plane for which the differential equation xy  y would have a unique solution through a point (x0, y0) in R. y

a) Verify that y  tan (x  c) is a one-parameter family of solutions of the differential equation y  1  y2 . (b) Since f(x, y)  1  y2 and fy  2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initial-value problem y  1  y2 , y(0)  0. Even though x0  0 is in the interval (2, 2), explain why the solution is not defined on this interval. (c) Determine the largest interval I of definition for the solution of the initial-value problem in part (b). 31

(a) Verify that y  1(x  c) is a one-parameter family of solutions of the differential equation y  y2 . (b) Since f(x, y)  y2 and fy  2y are continuous everywhere, the region R in Theorem 1.2.1 can be taken to be the entire xy-plane. Find a solution from the family in part (a) that satisfies y(0)  1. Then find a solution from the family in part (a) that satisfies y(0)  1. Determine the largest interval I of definition for the solution of each initial-value problem. (c) Determine the largest interval I of definition for the solution of the first-order initial-value problem y  y2 , y(0)  0. [Hint: The solution is not a member of the family of solutions in part (a).] 32.

(a) Show that a solution from the family in part (a) of Problem 31 that satisfies y  y2 , y(1)  1, is y  1(2  x). (b) Then show that a solution from the family in part (a) of Problem 31 that satisfies y  y2 , y(3)  1, is y  1(2  x). (c) Are the solutions in parts (a) and (b) the same? 33.

(a) Verify that 3x2  y2  c is a one-parameter family of solutions of the differential equation y dydx  3x. (b) By hand, sketch the graph of the implicit solution 3x2  y2  3. Find all explicit solutions y  (x) of the DE in part (a) defined by this relation. Give the interval I of definition of each explicit solution (c) The point (2, 3) is on the graph of 3x2  y2  3, but which of the explicit solutions in part (b) satisfies y(2)  3? 34

(a) Use the family of solutions in part (a) of Problem 33 to find an implicit solution of the initial-value(a) Use the family of solutions in part (a) of Problem 33to find an implicit solution of the initial-value

In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2  f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (

In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2  f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (

In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2  f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (

In Problems 3538 the graph of a member of a family of solutions of a second-order differential equation d2 ydx2  f(x, y, y) is given. Match the solution curve with at least one pair of the following initial conditions. (

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

In Problems 3944, is a twoparameter family of solutions of the second-order DE If possible, find a solution of the differential equation that satisfies the given side conditions. The conditions specified at two different points are called boundary conditions.

Find a function y  f(x) whose graph at each point (x, y) has the slope given by 8e2x  6x and has the y-intercept (0, 9)

Find a function y  f(x) whose second derivative is y  12x  2 at each point (x, y) on its graph and y  x  5 is tangent to the graph at the point corresponding to x  1.

Consider the initial-value problem y  x  2y, . Determine which of the two curves shown in Figure 1.2.11 is the only plausible solution curve. Explain your reasoning.

Determine a plausible value of x0 for which the graph of the solution of the initial-value problem y  2y  3x  6, y(x0)  0 is tangent to the x-axis at (x0, 0). Explain your reasoning.

Suppose that the first-order differential equation dydx  f(x, y) possesses a one-parameter family of solutions and that f(x, y) satisfies the hypotheses of Theorem 1.2.1 in some rectangular region R of the xy-plane. Explain why two different solution curves cannot intersect or be tangent to each other at a point (x0, y0) in R.

The functions and have the same domain but are clearly different. See Figures 1.2.12(a) and 1.2.12(b), respectively. Show that both functions are solutions of the initial-value problem dydx  xy1/2, y(2)  1 on the interval (, ). Resolve the apparent contradiction between this fact and the last sentence in Example 5.

Beginning in the next section we will see that differential equations can be used to describe or model many different physical systems. In this problem suppose that a model of the growing population of a small community is given by the initial-value problem where P is the number of individuals in the community and time t is measured in years. How fastthat is, at what rateis the population increasing at t  0? How fast is the population increasing when the population is 500?