Chapter 1

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

An RL circuit has EMF \(E(t)=10 \sin 4 t\) V. If \(R=\) \(2 \Omega, L=\frac{2}{3} \mathrm{H},\) and there is no current flowing initially, determine the current for \(t \geq 0\).

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. $$y(x)=c_{1} x^{1 / 2}, \quad y^{\prime}=\frac{y}{2 x}$$.

\((x-a)(x-b) y^{\prime}-(y-c)=0,\) where \(a, b, c\) are constants, with \(a \neq b.\)

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, \(c\) denotes an arbitrary constant.) $$y^{2}=c x, \quad 2 x d y-y d x=0, \quad y(1)=2$$

Find the equation of the orthogonal trajectories to the given family of curves. In each case, sketch some curves from each family. $$x^{2}+9 y^{2}=c$$

Sketch the slope field and some representative solution curves for the given differential equation. $$y^{\prime}=2 x y$$

Solve the given differential equation. $$\left(1+x^{2}\right) y^{\prime \prime}=-2 x y^{\prime}$$

Solve the given differential equation. $$(\sin y+y \cos x) d x+(x \cos y+\sin x) d y=0$$

Solve the given differential equation. $$\sin \left(\frac{y}{x}\right)\left(x y^{\prime}-y\right)=x \cos \left(\frac{y}{x}\right)$$

Solve the given differential equation. $$(1-y \sin x) d x-(\cos x) d y=0$$

Consider the RC circuit with \(R=2 \Omega, C=\frac{1}{8} \mathrm{F}\) and \(E(t)=10 \cos 3 t\) V. If \(q(0)=1 \mathrm{C},\) determine the current in the circuit for \(t \geq 0\).

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. $$y(x)=e^{-x} \sin 2 x, \quad y^{\prime \prime}+2 y^{\prime}+5 y=0$$.

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

Verify that the given function (or relation) defines a solution to the given differential equation and sketch some of the solution curves. If an initial condition is given, label the solution curve corresponding to the resulting unique solution. (In these problems, \(c\) denotes an arbitrary constant.) $$(x-c)^{2}+y^{2}=c^{2}, \quad y^{\prime}=\frac{y^{2}-x^{2}}{2 x y}, \quad y(2)=2$$

At time \(t\) the velocity, \(v(t),\) of an object is governed by the differential equation $$ \frac{d v}{d t}=\frac{1}{2}(25-v), \quad t>0 $$ (a) Verify that \(v(t)=25\) is a solution to this differential equation. (b) Sketch the slope field for \(0 \leq v \leq 25 .\) What happens to \(v(t)\) as \(t \rightarrow \infty ?\)

Solve the given differential equation. $$y^{\prime \prime}+y^{-1}\left(y^{\prime}\right)^{2}=y e^{-y}\left(y^{\prime}\right)^{3}$$

Solve the given differential equation. $$[1+\ln (x y)] d x+x y^{-1} d y=0$$

Solve the given differential equation. $$x y^{\prime}=\sqrt{16 x^{2}-y^{2}}+y, \quad x>0$$

Solve the given differential equation. $$y^{\prime}-x^{-1} y=2 x^{2} \ln x$$

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. $$y(x)=c_{1} \cosh 3 x+c_{2} \sinh 3 x, \quad y^{\prime \prime}-9 y=0$$.

Solve the given initial-value problem. \(\left(1-x^{2}\right) y^{\prime}+x y=a x, \quad y(0)=2 a,\) where \(a\) is a constant.

Prove that the initial-value problem $$ y^{\prime}=x \sin (x+y), \quad y(0)=1 $$ has a unique solution.

Consider the ontogenetic model $$ \frac{d m}{d t}=a m^{3 / 4}\left[1-\left(\frac{m}{M}\right)^{1 / 4}\right] $$ (a) Determine all equilibrium solutions. (b) Explain why \(\frac{d m}{d t}>0\) (c) Determine where in the region of physical interest the solution curves are concave up, and where they are concave down. (d) Determine the slope of the solution curves at the point of inflection. (e) Sketch the slope field and include some representative solution curves.

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. A population of swans in a wildlife sanctuary is declining due to the presence of dangerous chemicals in the water. If the population of swans is experiencing exponential decay, and if there were 400 swans in the park at the beginning of the summer and 340 swans 30 days later, (a) how many swans are in the park 60 days after the start of summer? 100 days after the start of summer? (b) how long does it take for the population of swans to be cut in half? (This is known as the half-life of the population.)

Solve the given differential equation. $$y^{\prime \prime}-y^{\prime} \tan x=1, \quad 0 \leq x<\pi / 2$$

Solve the given differential equation. $$x^{-1}(x y-1) d x+y^{-1}(x y+1) d y=0$$

Solve the given differential equation. $$x y^{\prime}-y=\sqrt{9 x^{2}+y^{2}}, \quad x>0$$

Solve the given differential equation. \(y^{\prime}+\alpha y=e^{\beta x},\) where \(\alpha, \beta\) are constants.

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. $$y(x)=c_{1} x^{-3}+c_{2} x^{-1}, \quad x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0$$.

Solve the given initial-value problem. $$\frac{d y}{d x}=1-\frac{\sin (x+y)}{\sin y \cos x}, \quad y(\pi / 4)=\pi / 4$$

Use the existence and uniqueness theorem to prove that \(y(x)=3\) is the only solution to the initial-value problem $$ y^{\prime}=\frac{x}{x^{2}+1}\left(y^{2}-9\right), \quad y(0)=3 $$

An object of mass \(m\) is released from rest in a medium in which the frictional forces are proportional to the square of the velocity. The initial- value problem that governs the subsequent motion is $$m v \frac{d v}{d y}=m g-k v^{2}, \quad v(0)=0$$ where \(v(t)\) denotes the velocity of the object at time \(t, y(t)\) denotes the distance travelled by the object at time \(t\) as measured from the point at which the object was released, and \(k\) is a positive constant.(a) Solve \((1.12 .6)\) and show that $$ v^{2}=\frac{m g}{k}\left(1-e^{-2 k y / m}\right) $$ (b) Make a sketch of \(v^{2}\) as a function of \(y\)

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. At the conclusion of the Super Bowl, the number of fans remaining in the stadium decreases at a rate proportional to the number of fans in the stadium. Assume that there are 100,000 fans in the stadium at the end of the Super Bowl and ten minutes later there are 80,000 fans in the stadium. (a) Thirty minutes after the Super Bowl will there be more or less than 40,000 fans? How do you know this without doing any calculations? (b) What is the half-life (see the previous problem) for the fan population in the stadium? (c) When will there be only 15,000 fans left in the stadium? (d) Explain why the exponential decay model for the population of fans in the stadium is not realistic from a qualitative perspective.

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

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

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

Solve the given differential equation. \(y^{\prime}+m x^{-1} y=\ln x,\) where \(m\) is constant.

Determine the current flowing in an RL circuit if the applied EMF is \(E(t)=E_{0} \sin \omega t,\) where \(E_{0}\) and \(\omega\) are constants. Identify the transient part of the solution and the steady-state solution.

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. $$y(x)=c_{1} x^{2} \ln x, \quad x^{2} y^{\prime \prime}-3 x y^{\prime}+4 y=0$$.

Solve the given initial-value problem. $$y^{\prime}=y^{3} \sin x, \quad y(0)=0$$

Do you think that the initial-value problem $$ y^{\prime}=x y^{1 / 2}, \quad y(0)=0 $$ has a unique solution? Justify your answer.

Solve the given differential equation. $$x y^{\prime}+y \ln x=y \ln y$$

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. Cobalt-60, an isotope used in cancer therapy, decays exponentially with a half-life of 5.2 years (i.e., half the original sample remains after 5.2 years). How long does it take for a sample of Cobalt-60 to disintegrate to the extent that only \(4 \%\) of the original amount remains?

Solve the given initial-value problem. \(y^{\prime \prime}=\omega^{2} y, \quad y(0)=a, \quad y^{\prime}(0)=0,\) where \(\omega, a\) are positive constants.

\diamond Use the Fourth-Order Runge-Kutta Method with \(h=0.5\) to approximate the solution to the initial-value problem $$y^{\prime}+\frac{1}{10} y=e^{-x / 10} \cos x, \quad y(0)=0$$ at the points \(x=0.5,1.0, \ldots, 25 .\) Plot these points and describe the behavior of the corresponding solution.

Solve the given initial-value problem. $$2 x^{2} y^{\prime}+4 x y=3 \sin x, \quad y(2 \pi)=0$$

Solve the given initial-value problem. $$y^{\prime}+2 x^{-1} y=4 x, \quad y(1)=2$$

One solution to the initial-value problem $$ \frac{d y}{d x}=\frac{2}{3}(y-1)^{1 / 2}, \quad y(1)=1 $$ is \(y(x)=1 .\) Determine another solution to this initialvalue problem. Does this contradict the existence and uniqueness theorem (Theorem 1.3 .2 )? Explain.

Even simple looking differential equations can have complicated solution curves. In this problem, we study the solution curves of the differential equation $$ y^{\prime}=-2 x y^{2} $$ (a) Verify that the hypotheses of the existence and uniqueness theorem (Theorem 1.3 .2 ) are satisfied for the initial-value problem $$ y^{\prime}=-2 x y^{2}, \quad y\left(x_{0}\right)=y_{0} $$ for every \(\left(x_{0}, y_{0}\right) .\) This establishes that the initialvalue problem always has a unique solution on some interval containing \(x_{0}\) (b) Verify that for all values of the constant \(c, y(x)=\) \(\frac{1}{\left(x^{2}+c\right)}\) is a solution to \((1.3 .8)\) (c) Use the solution to \((1.3 .8)\) given in (b) to solve the following initial-value problem. For each case, sketch the corresponding solution curve, and state the maximum interval on which your solution is valid. (i) \(y^{\prime}=-2 x y^{2}, \quad y(0)=1\) (ii) \(y^{\prime}=-2 x y^{2}, \quad y(1)=1\) (iii) \(y^{\prime}=-2 x y^{2}, \quad y(0)=-1\) (d) What is the unique solution to the initial-value problem \(y^{\prime}=-2 x y^{2}, \quad y(0)=0 ?\)