Chapter 1

Solve the given differential equation. $$\frac{d y}{d x}=\frac{y^{2}+2 x y-2 x^{2}}{x^{2}-x y+y^{2}}$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\frac{d y}{d x}=\frac{2 \ln x}{x y}$$

The following initial-value problem arises in the analysis of a cable suspended between two fixed points $$y^{\prime \prime}=\frac{1}{a} \sqrt{1+\left(y^{\prime}\right)^{2}}, \quad y(0)=a, \quad y^{\prime}(0)=0$$, where \(a\) is a nonzero constant. Solve this initial-value problem for \(y(x) .\) The corresponding solution curve is called a catenary.

Solve the given initial-value problem. $$\left(3 x^{2} \ln x+x^{2}-y\right) d x-x d y=0, \quad y(1)=5$$

Solve the given initial-value problem. $$(\sin x) y^{\prime}-y \cos x=\sin 2 x, \quad y(\pi / 2)=2$$

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} x^{1 / 2}+3 x^{2}, \quad 2 x^{2} y^{\prime \prime}-x y^{\prime}+y=9 x^{2}$$.

Consider the initial-value problem: $$ y^{\prime}=y(y-1), \quad y\left(x_{0}\right)=y_{0} $$ (a) Verify that the hypotheses of the existence and uniqueness theorem are satisfied for this initialvalue problem for any \(x_{0}, y_{0} .\) This establishes that the initial-value problem always has a unique solution on some interval containing \(x_{0}\) (b) By inspection, determine all equilibrium solutions to the differential equation. (c) Determine the regions in the \(x y\) -plane where the solution curves are concave up, and determine those regions where they are concave down. (d) Sketch the slope field for the differential equation, and determine all values of \(y_{0}\) for which the initial-value problem has bounded solutions. On your slope field, sketch representative solution curves in the three cases \(y_{0}<0,01\)

Solve the given differential equation. $$2 x y d y-\left(x^{2} e^{-y^{2} / x^{2}}+2 y^{2}\right) d x=0$$

According to data from the U.S. Bureau of the Census, the population (measured in millions of people)of the U.S. in 1950, 1960, and 1970 was, respectively, 151.3, 179.4, and 203.3. (a) Using the 1950 and 1960 population figures, solve the corresponding Malthusian population model. (b) Determine the logistic model corresponding to the given data. (c) On the same set of axes, plot the solution curves obtained in (a) and (b). From your plots, determine the values the different models would have predicted for the population in 1980 and 1990 , and compare these predictions to the actual values of 226.54 and 248.71 , respectively.

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$x y^{\prime}-2 y=2 x^{2} \ln x$$

Consider the general second-order linear differential equation with dependent variable missing: $$y^{\prime \prime}+p(x) y^{\prime}=q(x)$$ Replace this differential equation with an equivalent pair of first-order equations and express the solution in terms of integrals.

Solve the given initial-value problem. $$\left(y e^{x y}+\cos x\right) d x+x e^{x y} d y=0, y(\pi / 2)=0$$

Solve the given initial-value problem. $$\frac{d x}{d t}+\frac{2}{4-t} x=5, x(0)=4$$

Consider the special case of the RLC circuit in which the resistance is negligible and the driving EMF is zero. The differential equation governing the charge on the capacitor in this case is $$ \frac{d^{2} q}{d t^{2}}+\frac{1}{L C} q=0 $$ If the capacitor has an initial charge of \(q_{0}\) coulombs, and there is no current flowing initially, determine the charge on the capacitor for \(t>0,\) and the corresponding current in the circuit. [Hint: Let \(u=\frac{d q}{d t}\) and use the chain rule to show that this implies \(\frac{d u}{d t}=\) \(\left.u\left(\frac{d u}{d q}\right) \cdot\right]\)

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$\begin{array}{l} y(x)=c_{1} x^{2}+c_{2} x^{3}-x^{2} \sin x. \\ x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x^{4} \sin x. \end{array}$$

Find the equation of the curve that passes through the point \((0,1 / 2)\) and whose slope at each point \((x, y)\) is \(-\frac{x}{4 y}.\)

For Problems (a) Determine all equilibrium solutions. (b) Determine the regions in the \(x y\) -plane where the solutions are increasing, and determine those regions where they are decreasing. (c) Determine the regions in the \(x y\) -plane where the solution curves are concave up, and determine those regions where they are concave down. (d) Sketch representative solution curves in each region of the \(x y\) -plane identified in (c). $$y^{\prime}=(y+2)(y-1)$$

Solve the given differential equation. $$x^{2} \frac{d y}{d x}=y^{2}+3 x y+x^{2}$$

In a period of five years, the population of a city doubles from its initial size of 50 (measured in thousands of people). After ten more years, the population has reached 250. Determine the logistic model corresponding to this data. Sketch the solution curve and use your plot to estimate the time it will take for the population to reach 95% of the carrying capacity.

Consider the general third-order differential equation of the form $$y^{\prime \prime \prime}=F\left(x, y^{\prime \prime}\right)$$ $$(1.11 .27)$$ (a) Show that Equation \((1.11 .27)\) can be replaced by the equivalent first- order system $$\frac{d u_{1}}{d x}=u_{2}, \quad \frac{d u_{2}}{d x}=u_{3}, \quad \frac{d u_{3}}{d x}=F\left(x, u_{3}\right)$$, where the variables \(u_{1}, u_{2}, u_{3}\) are defined by $$u_{1}=y, \quad u_{2}=y^{\prime}, \quad u_{3}=y^{\prime \prime}$$. $$\text { (b) Solve } y^{\prime \prime \prime}=x^{-1}\left(y^{\prime \prime}-1\right)$$

Show that if \(\phi(x, y)\) is a potential function for \(M(x, y) d x+N(x, y) d y=0,\) then so is \(\phi(x, y)+c\) where \(c\) is an arbitrary constant. This shows that potential functions are only uniquely defined up to an additive constant.

Solve the given initial-value problem. $$\left(y-e^{x}\right) d x+d y=0, \quad y(0)=1$$

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. \(y(x)=c_{1} e^{a x}+c_{2} e^{b x}, \quad y^{\prime \prime}-(a+b) y^{\prime}+a b y=0\) where \(a\) and \(b\) are constants and \(a \neq b\)

Find the equation of the curve that passes through the point (3,1) and whose slope at each point \((x, y)\) is \(e^{x-y}.\)

\(m\) denotes a fixed nonzero constant, and \(c\) is the constant distinguishing the different curves in the given family. In each case, find the equation of the orthogonal trajectories. $$y^{2}=m x+c$$

Solve the given differential equation. $$y y^{\prime}=\sqrt{x^{2}+y^{2}}-x, \quad x>0$$

A simple pendulum consists of a particle of mass \(m\) supported by a piece of string of length \(L .\) Assuming that the pendulum is displaced through an angle \(\theta_{0}\) radians from the vertical and then released from rest, the resulting motion is described by the initial-value problem $$\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \sin \theta=0, \quad \theta(0)=\theta_{0}, \quad \frac{d \theta}{d t}(0)=0$$ $$(1.11 .28)$$ (a) For small oscillations, \(\theta<<1,\) we can use the approximation \(\sin \theta \approx \theta\) in Equation \((1.11 .28)\) to obtain the linear equation $$\frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \theta=0, \quad \theta(0)=\theta_{0}, \quad \frac{d \theta}{d t}(0)=0$$. Solve this initial- value problem for \(\theta\) as a function of \(t .\) Is the predicted motion reasonable? (b) Obtain the following first integral of \((1.11 .28):\) $$\frac{d \theta}{d t}=\pm \sqrt{\frac{2 g}{L}\left(\cos \theta-\cos \theta_{0}\right)}$$. $$(1.11 .29)$$ (c) Show from Equation \((1.11 .29)\) that the time \(T\) (equal to one-fourth of the period of motion) required for \(\theta\) to go from 0 to \(\theta_{0}\) is given by the elliptic integral of the first kind $$T=\sqrt{\frac{L}{2 g}} \int_{0}^{\theta_{0}} \frac{1}{\sqrt{\cos \theta-\cos \theta_{0}}} d \theta$$. (d) Show that \((1.11 .30)\) can be written as $$T=\sqrt{\frac{L}{g}} \int_{0}^{\pi / 2} \frac{1}{\sqrt{1-k^{2} \sin ^{2} u}} d u$$ $$\begin{aligned} &\text { where } k=\sin \left(\theta_{0} / 2\right) . \text { [Hint: First express } \cos \theta\\\ &\text { and }\left.\cos \theta_{0} \text { in terms of } \sin ^{2}(\theta / 2) \text { and } \sin ^{2}\left(\theta_{0} / 2\right) .\right] \end{aligned}$$

Determine whether the given function is an integrating factor for the given differential equation. $$I(x, y)=\cos (x y),[\tan (x y)+x y] d x+x^{2} d y=0$$

Solve the given initial-value problem. \(y^{\prime}+y=f(x), \quad y(0)=3,\) where $$f(x)=\left\\{\begin{array}{ll} 1, & \text { if } x \leq 1 \\ 0, & \text { if } x > 1 \end{array}\right.$$

Find the equation of the curve that passes through the point (-1,1) and whose slope at each point \((x, y)\) is \(x^{2} y^{2}.\)

Solve the given differential equation. $$2 x(y+2 x) y^{\prime}=y(4 x-y)$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$y^{\prime}+y(\tan x+y \sin x)=0$$

Determine whether the given function is an integrating factor for the given differential equation. $$I(x, y)=y^{-2} e^{-x / y}, y\left(x^{2}-2 x y\right) d x-x^{3} d y=0$$

Solve the given initial-value problem. $$\begin{aligned} y^{\prime}-2 y=f(x), & y(0)=1, \text { where } \\ f(x) &=\left\\{\begin{array}{ll} 1-x, & \text { if } x < 1 \\ 0, & \text { if } x \geq 1 \end{array}\right. \end{aligned}$$

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. \(y(x)=e^{a x}\left(c_{1} \cos b x+c_{2} \sin b x\right)\) \(y^{\prime \prime}-2 a y^{\prime}+\left(a^{2}+b^{2}\right) y=0,\) where \(a\) and \(b\) are constants.

At time \(t,\) the velocity \(v(t)\) of an object moving in a straight line satisfies $$ \frac{d}{d x}-\left(4+y^{2}\right) $$ (a) Show that $$ \tan ^{-1}(v)=\tan ^{-1}\left(v_{0}\right)-t $$ where \(v_{0}\) denotes the velocity of the object at time \(t=0\) (and we assume \(v_{0}>0\) ). Hence prove that the object comes to rest after a finite time \(\tan ^{-1}\left(v_{0}\right) .\) Does the object remain at rest? (b) Use the chain rule to show that \((1.4 .22)\) can be written as \(v \frac{d v}{d x}=-\left(1+v^{2}\right),\) where \(x(t)\) denotes the distance travelled by the object at time \(t,\) from its position at \(t=0 .\) Determine the distance traveelled by the object when it first comes to rest.

For Problems (a) Determine all equilibrium solutions. (b) Determine the regions in the \(x y\) -plane where the solutions are increasing, and determine those regions where they are decreasing. (c) Determine the regions in the \(x y\) -plane where the solution curves are concave up, and determine those regions where they are concave down. (d) Sketch representative solution curves in each region of the \(x y\) -plane identified in (c) $$y^{\prime}=y(y-1)(y+1)$$

Solve the given differential equation. $$x \frac{d y}{d x}=x \tan (y / x)+y$$

Determine whether the given function is an integrating factor for the given differential equation. $$I(x)=\sec x,\left[2 x-\left(x^{2}+y^{2}\right) \tan x\right] d x+2 y d y=0$$

Determine all values of the constant \(r\) such that the given function solves the given differential equation. $$y(x)=e^{r x}, \quad y^{\prime \prime}-y^{\prime}-6 y=0$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=4 x$$

We call a coordinate system \((u, v)\) orthogonal if its coordinate curves (the two families of curves \(u=\) constant and \(v=\) constant ) are orthogonal trajectories (for example, a Cartesian coordinate system or a polar coordinate system). Let \((u, v)\) be orthogonal coordinates, where \(u=x^{2}+2 y^{2},\) and \(x\) and \(y\) are Cartesian coordinates. Find the Cartesian equation of the \(v\) -coordinate curves, and sketch the \((u, v)\) coordinate system.

Solve the given differential equation. $$\frac{d y}{d x}=\frac{x \sqrt{x^{2}+y^{2}}+y^{2}}{x y}, \quad x>0$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$\left(y-x^{2}\right) d x+2 x d y=0, \quad x>0$$

Find the general solution to the second-order differential equation $$\frac{d^{2} y}{d x^{2}}+\frac{1}{x} \frac{d y}{d x}=9 x, \quad x > 0$$ [Hint: Let \(\left.u=\frac{d y}{d x} .\right]\)

The pressure \(p,\) and density, \(\rho,\) of the atmosphere at a height \(y\) above the earth's surface are related by $$ d p=-g \rho d y $$ Assuming that \(p\) and \(\rho\) satisfy the adiabatic equation of state \(p=p_{0}\left(\frac{\rho}{\rho_{0}}\right)^{\gamma},\) where \(\gamma \neq 1\) is a constant and \(p_{0}\) and \(\rho_{0}\) denote the pressure and density at the earth's surface, respectively, show that $$ p=p_{0}\left[1-\frac{(\gamma-1)}{\gamma} \cdot \frac{\rho_{0} g y}{p_{0}}\right]^{\gamma /(\gamma-1)} $$.

Determine all values of the constant \(r\) such that the given function solves the given differential equation. $$y(x)=e^{r x}, \quad y^{\prime \prime}+6 y^{\prime}+9 y=0$$.

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=1 / x$$

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\frac{d y}{d x}=\frac{\sin y+y \cos x+1}{1-x \cos y-\sin x}$$

Determine an integrating factor for the given differential equation, and hence find the general solution. $$\left(3 x y-2 y^{-1}\right) d x+x\left(x+y^{-2}\right) d y=0$$