1–22 solve the given differential equation by

(a) Use a CAS and the concept of level curves to plot representative graphs of members of the family of solutions of the differential equation \(\frac{d y}{d x}=\frac{x(1-x)}{y(-2+y)}\). Experiment with different numbers of level curves as well as various rectangular regions in the xy-plane until your result resembles Figure 2.2.6. (b) On separate coordinate axes, plot the graph of the implicit solution corresponding to the initial condition \(y(0)=\frac{3}{2}\). Use a colored pencil to mark off that segment of the graph that corresponds to the solution curve of a solution \(\phi\) that satisfies the initial condition. With the aid of a root-finding application of a CAS, determine the approximate largest interval I of definition of the solution \(\phi\). [Hint: First find the points on the curve in part (a) where the tangent is vertical.] (c) Repeat part (b) for the initial condition y(0) = -2. Text Transcription: dy/dx = x(1-x) / y(-2+y) y(0)=3/2 phi

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=(x+1)^{2}\) Text Transcription: dy/dx=(x+1)^2

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=\sin 5 x\) Text Transcription: dy/dx = sin 5x

In Problems 1–22 solve the given differential equation by separation of variables. \(d x+e^{3 x} d y=0\) Text Transcription: dx+e^3x dy=0

In Problems 1–22 solve the given differential equation by separation of variables. \(x \frac{d y}{d x}=4 y\) Text Transcription: x dy/dx=4y

In Problems 1–22 solve the given differential equation by separation of variables. \(d y-(y-1)^{2} d x=0\) Text Transcription: dy-(y-1)^2 dx=0

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}+2 x y^{2}=0\) Text Transcription: dy/dx + 2xy^2=0

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=e^{3 x+2 y}\) Text Transcription: dy/dx=e^3x+2y

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2}\) Text Transcription: dy/dx=(2y+3/4x+5)^2

In Problems 1–22 solve the given differential equation by separation of variables. \(e^{x} y \frac{d y}{d x}=e^{-y}+e^{-2 x-y}\) Text Transcription: e^x y dy/dx = e^-y + e^-2x-y

In Problems 1–22 solve the given differential equation by separation of variables. \(\sin 3 x \ d x+2 y \cos ^{3} 3 x \ d y=0\) Text Transcription: sin 3xdx + 2y cos^3 3xdy=0

In Problems 1–22 solve the given differential equation by separation of variables. \(x\left(1+y^{2}\right)^{1 / 2} \ d x=y\left(1+x^{2}\right)^{1 / 2} \ d y\) Text Transcription: x(1+y^2)^1/2 dx = y(1+x^2)^1/2 dy

In Problems 1–22 solve the given differential equation by separation of variables. \(\csc y \ d x+\sec ^{2} x \ d y=0\) Text Transcription: csc y dx+sec^2 x dy=0

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d S}{d r}=k S\) Text Transcription: dS/dr=kS

In Problems 1–22 solve the given differential equation by separation of variables. \(\left(e^{y}+1\right)^{2} e^{-y} \ d x+\left(e^{x}+1\right)^{3} e^{-x} \ d y=0\) Text Transcription: (e^y+1)^2 e^-ydx+(e^x+1)^3 e^-xdy=0

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d Q}{d t}=k(Q-70)\) Text Transcription: dQ/dt=k(Q-70)

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d P}{d t}=P-P^{2}\) Text Transcription: dP/dt=P-P^2

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d N}{d t}+N=N t e^{t+2}\) Text Transcription: dN/dt+N=Nte^t+2

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8}\) Text Transcription: dy/dx = xy+3x-y-3 / xy-2x+4y-8

In Problems 23–28 find an explicit solution of the given initial-value problem. \(x^{2} \frac{d y}{d x}=y-x y\), y(-1)=-1 Text Transcription: x^2 dy/dx = y-xy

In Problems 1–22 solve the given differential equation by separation of variables. \(\left(e^{x}+e^{-x}\right) \ \frac{d y}{d x}=y^{2}\) Text Transcription: (e^x + e^-x) dy/dx = y^2

In Problems 23–28 find an explicit solution of the given initial-value problem. \(\frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}\), y(2)=2 Text Transcription: dy/dx = y^2 - 1 / x^2 - 1

In Problems 23–28 find an explicit solution of the given initial-value problem. \(\frac{d x}{d t}=4\left(x^{2}+1\right)\), \(x(\pi/ 4)=1\) Text Transcription: dx/dt = 4(x^2 + 1) x(pi/4) = 1

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3}\) Text Transcription: dy/dx = xy+2y-x-2 / xy-3y+x-3

In Problems 1–22 solve the given differential equation by separation of variables. \(\frac{d y}{d x}=x \sqrt{1-y^{2}}\) Text Transcription: dy/dx=x sqrt 1-y^2

In Problems 23–28 find an explicit solution of the given initial-value problem. \(\frac{d y}{d t}+2 y=1\), \(y(0)=\frac{5}{2}\) Text Transcription: dy/dt + 2y=1 y(0)=5/2

In Problems 23–28 find an explicit solution of the given initial-value problem. \(\sqrt{1-y^{2}} \ d x-\sqrt{1-x^{2}} \ d y=0\), \(y(0)=\frac{\sqrt{3}}{2}\) Text Transcription: 1-y^2 dx - 1-x^2 dy=0 y(0)=sqrt 3 / 2

In Problems 23–28 find an explicit solution of the given initial-value problem. \(\left(1+x^{4}\right) \ d y+x\left(1+4 y^{2}\right) \ d x=0\), y(1)=0 Text Transcription: (1+x^4) dy + x(1+4y^2) dx=0

In Problems 29 and 30 proceed as in Example 5 and find an explicit solution of the given initial-value problem. \(\frac{d y}{d x}=y e^{-x^{2}}\), y(4)=1 Text Transcription: dy/dx=ye^-x^2

In Problems 31–34 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. \((2 y-2) \frac{d y}{d x}=3 x^{2}+4 x+2\), y(1)=-2 Text Transcription: (2y-2) dy/dx = 3x^2 + 4x + 2

In Problems 29 and 30 proceed as in Example 5 and find an explicit solution of the given initial-value problem. \(\frac{d y}{d x}=y^{2} \sin x^{2}\), \(y(-2)=\frac{1}{3}\) Text Transcription: dy/dx = y^2 sin x^2 y(-2)=1/3

In Problems 31–34 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. \(\frac{d y}{d x}=\frac{2 x+1}{2 y}\), y(-2)=-1 Text Transcription: dy/dx = 2x+1/2y

In Problems 31–34 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. \(e^{y} \ d x-e^{-x} \ d y=0\), y(0)=0 Text Transcription: e^y dx - e^-x dy = 0

In Problems 31–34 find an explicit solution of the given initial-value problem. Determine the exact interval I of definition by analytical methods. Use a graphing utility to plot the graph of the solution. sin x dx + y dy = 0, y(0) = 1

Find a singular solution of Problem 21. Of Problem 22.

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39–42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). \(\frac{d y}{d x}=(y-1)^{2}\), y(0)=1 Text Transcription: dy/dx=(y-1)^2

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39–42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). \(\frac{d y}{d x}=(y-1)^{2}\), y(0)=1.01 Text Transcription: dy/dx=(y-1)^2

(a) Find a solution of the initial-value problem consisting of the differential equation in Example 3 and each of the initial-conditions: y(0) = 2, y(0) = -2, and \(y\left(\frac{1}{4}\right)=1\) . (b) Find the solution of the differential equation in Example 4 when ln \(\mathcal{c}_{1}\) is used as the constant of integration on the left-hand side in the solution and \(4 \ln c_{1}\) is replaced by ln c. Then solve the same initial-value problems in part (a). Text Transcription: y(1/4)=1 c_1 4 ln c_1

Find a solution of \(x \frac{d y}{d x}=y^{2}-y\) that passes through the indicated points. (a) (0, 1) (b) (0, 0) (c) \(\left(\frac{1}{2}, \frac{1}{2}\right)\) (d) \(\left(2, \frac{1}{4}\right)\) Text Transcription: x dy/dx = y^2 - y (1/2, 1/2) (2, 1/4)

Show that an implicit solution of \(2 x \sin ^{2} y \ d x-\left(x^{2}+10\right) \cos y \ d y=0\) is given by \(\ln \left(x^{2}+10\right)+\csc y=c\). Find the constant solutions, if any, that were lost in the solution of the differential equation. Text Transcription: 2x sin^2 y dx - (x^2 + 10) cos y dy = 0 ln (x^2 + 10) + csc y = c

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39–42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). \(\frac{d y}{d x}=(y-1)^{2}+0.01\), y(0)=1 Text Transcription: dy/dx = (y-1)^2 + 0.01

Often a radical change in the form of the solution of a differential equation corresponds to a very small change in either the initial condition or the equation itself. In Problems 39–42 find an explicit solution of the given initial-value problem. Use a graphing utility to plot the graph of each solution. Compare each solution curve in a neighborhood of (0, 1). \(\frac{d y}{d x}=(y-1)^{2}-0.01\), y(0)=1 Text Transcription: dy/dx = (y-1)^2 - 0.01

Every autonomous first-order equation dy/dx = f(y) is separable. Find explicit solutions \(y_{1}(x)\), \(y_{2}(x)\), \(y_{3}(x)\), and \(y_{4}(x)\) of the differential equation \(d y / d x=y-y^{3}\) that satisfy, in turn, the initial conditions \(y_{1}(0)=2\), \(y_{2}(0)=\frac{1}{2}\), \(y_{3}(0)=-\frac{1}{2}\) and \(y_{4}(0)=-2\). Use a graphing utility to plot the graphs of each solution. Compare these graphs with those predicted in Problem 19 of Exercises 2.1. Give the exact interval of definition for each solution. Text Transcription: y_1(x) y_2(x) y_3(x) y_4(x) dy/dx=y-y^3 y_1(0)=2 y_2(0)=1/2 y_3(0)=-1/2 y_4(0)=-2

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \(\frac{d y}{d x}=\frac{1}{1+\sin x}\) Text Transcription: dy/dx=1 / 1+sin x

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \((\sqrt{x}+x) \frac{d y}{d x}=\sqrt{y}+y\) Text Transcription: (sqrt x + x) dy/dx = sqrt y + y

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \(\frac{d y}{d x}=\frac{\sin \sqrt{x}}{\sqrt{y}}\) Text Transcription: dy/dx=sin sqrt x / sqrt y

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \(\frac{d y}{d x}=y^{2 / 3}-y\) Text Transcription: dy/dx = y^2/3 - y

(a) The autonomous first-order differential equation dy/dx = 1/(y - 3) has no critical points. Nevertheless, place 3 on the phase line and obtain a phase portrait of the equation. Compute \(d^{2} y / d x^{2}\) to determine where solution curves are concave up and where they are concave down (see Problems 35 and 36 in Exercises 2.1). Use the phase portrait and concavity to sketch, by hand, some typical solution curves. (b) Find explicit solutions \(y_{1}(x)\), \(y_{2}(x)\), \(y_{3}(x)\), and \(y_{4}(x)\) of the differential equation in part (a) that satisfy, in turn, the initial conditions \(y_{1}(0)=4\), \(y_{2}(0)=2\), \(y_{3}(1)=2\), and \(y_{4}(-1)=4\). Graph each solution and compare with your sketches in part (a). Give the exact interval of definition for each solution. Text Transcription: d^2 y/dx^2 y_1(x) y_2(x) y_3(x) y_4(x) y_1(0)=4 y_2(0)=2 y_3(1)=2 y_4(-1)=4

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \(\frac{d y}{d x}=\frac{e^{\sqrt{x}}}{y}\), y(1)=4 Text Transcription: dy/dx=e^x / y

In Problems 45–50 use a technique of integration or a substitution to find an explicit solution of the given differential equation or initial-value problem. \(\frac{d y}{d x}=\frac{x \tan ^{-1} x}{y}\), y(0)=3 Text Transcription: dy/dx=x tan^-1 x/y

(a) Solve the two initial-value problems: \(\frac{d y}{d x}=y\), y(0)=1 and \(\frac{d y}{d x}=y+\frac{y}{x \ln x}\), y(e)=1 (b) Show that there are more than 1.65 million digits in the y-coordinate of the point of intersection of the two solution curves in part (a). Text Transcription: dy/dx = y dy/dx = y + y/x ln x

Find a function whose square plus the square of its derivative is 1.

(a) Explain why the interval of definition of the explicit solution \(y=\phi_{2}(x)\) of the initial-value problem in Example 2 is the open interval (-5, 5). (b) Can any solution of the differential equation cross the x-axis? Do you think that \(x^{2}+y^{2}=1\) is an implicit solution of the initial-value problem dy/dx = -x/y, y(1) = 0? Text Transcription: y=phi_2(x) x^2 + y^2 = 1

(a) If a > 0, discuss the differences, if any, between the solutions of the initial-value problems consisting of the differential equation dy/dx = x/y and each of the initial conditions y(a) = a, y(a) = -a, y(-a) = a, and y(-a) = -a. (b) Does the initial-value problem dy/dx = x/y, y(0) = 0 have a solution? (c) Solve dy/dx = x/y, y(1) = 2 and give the exact interval I of definition of its solution

In Problems 43 and 44 we saw that every autonomous first-order differential equation dy/dx = f(y) is separable. Does this fact help in the solution of the initial-value problem \(\frac{d y}{d x}=\sqrt{1+y^{2}} \sin ^{2} y\), \(y(0)=\frac{1}{2}\) ? Discuss. Sketch, by hand, a plausible solution curve of the problem. Text Transcription: dy/dx = sqrt 1 + y^2 sin^2 y y(0)=1/2

(a) The differential equation in Problem 27 is equivalent to the normal form \(\frac{d y}{d x}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}\) in the square region in the xy-plane defined by |x| < 1, |y| < 1. But the quantity under the radical is nonnegative also in the regions defined by |x| > 1, |y| > 1. Sketch all regions in the xy-plane for which this differential equation possesses real solutions. (b) Solve the DE in part (a) in the regions defined by |x| > 1, |y| > 1. Then find an implicit and an explicit solution of the differential equation subject to y(2) = 2. Text Transcription: dy/dx = sqrt 1-y^2/1-x^2

Suspension Bridge In (16) of Section 1.3 we saw that a mathematical model for the shape of a flexible cable strung between two vertical supports is \(\frac{d y}{d x}=\frac{W}{T_{1}}\) where W denotes the portion of the total vertical load between the points \(P_{1}\) and \(P_{2}\) shown in Figure 1.3.7. The DE (10) is separable under the following conditions that describe a suspension bridge. Let us assume that the x- and y-axes are as shown in Figure 2.2.5—that is, the x-axis runs along the horizontal roadbed, and the y-axis passes through (0, a), which is the lowest point on one cable over the span of the bridge, coinciding with the interval [-L/2, L/2]. In the case of a suspension bridge, the usual assumption is that the vertical load in (10) is only a uniform roadbed distributed along the horizontal axis. In other words, it is assumed that the weight of all cables is negligible in comparison to the weight of the roadbed and that the weight per unit length of the roadbed (say, pounds per horizontal foot) is a constant \(\rho\). Use this information to set up and solve an appropriate initial-value problem from which the shape (a curve with equation \(y=\phi(x)\)) of each of the two cables in a suspension bridge is determined. Express your solution of the IVP in terms of the sag h and span L. See Figure 2.2.5. Text Transcription: dy/dx=W/T_1 P_1 P_2 rho y=phi(x)

(a) Use a CAS and the concept of level curves to plot representative graphs of members of the family of solutions of the differential equation \(\frac{d y}{d x}=-\frac{8 x+5}{3 y^{2}+1}\). Experiment with different numbers of level curves as well as various rectangular regions defined by \(a \leq x \leq b\), \(c \leq y \leq d\). (b) On separate coordinate axes plot the graphs of the particular solutions corresponding to the initial conditions: y(0) = -1; y(0) = 2; y(-1) = 4; y(-1) = -3. Text Transcription: dy/dx = -8x + 5/3y^2 + 1 a leq x leq b c leq y leq d

(a) Find an implicit solution of the IVP \((2 y+2) \ d y-\left(4 x^{3}+6 x\right) \ d x=0\), y(0)=-3 (b) Use part (a) to find an explicit solution \(y=\phi(x)\) of the IVP. (c) Consider your answer to part (b) as a function only. Use a graphing utility or a CAS to graph this function, and then use the graph to estimate its domain. (d) With the aid of a root-finding application of a CAS, determine the approximate largest interval I of definition of the solution \(y=\phi(x)\) in part (b). Use a graphing utility or a CAS to graph the solution curve for the IVP on this interval. Text Transcription: (2y+2) dy - (4x^3 + 6x) dx = 0 y=phi(x)

In Problems 1–22 solve the given differential equation by separation of variables. \(y \ln x \frac{d x}{d y}=\left(\frac{y+1}{x}\right)^{2}\) Text Transcription: y ln x dx/dy=(y+1/x)^2