Chapter 1

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ (1-x) y^{\prime \prime}-4 x y^{\prime}+5 y=\cos x $$

The population model given in (1) fails to take death into consideration; the growth rate equals the birth rate. In another model of a changing population of a community, it is assumed that the rate at which the population changes is a net rate - that is, the difference between the rate of births and the rate of deaths in the community. Determine a model for the population \(P(t)\) if both the birth rate and the death rate are proportional to the population present at time \(t\).

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ x \frac{d^{3} y}{d x^{3}}-\left(\frac{d y}{d x}\right)^{4}+y=0 $$

y=1 /\left(1+c_{1} e^{-x}\right)\( is a one-parameter family of solutions of the first-order DE \)y^{\prime}=y-y^{2}$. Find a solution of the first-order IVP consisting of this differential equation and the given initial condition. $$ y(-1)=2 $$

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ t^{5} y^{(4)}-t^{3} y^{\prime \prime}+6 y=0 $$

\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0\). Find a 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. $$ y(2)=\frac{1}{3} $$

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ \frac{d^{2} u}{d r^{2}}+\frac{d u}{d r}+u=\cos (r+u) $$

\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0\). Find a 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. $$ y(-2)=\frac{1}{2} $$

In Problems 3 and 4 , fill in the blank and then write this result as a linear second-order differential equation that is free of the symbols \(c_{1}\) and \(c_{2}\) and has the form \(F\left(y, y^{\prime \prime}\right)=0\). The symbols \(c_{1}, c_{2}\), and \(k\) represent constants. \(\frac{d^{2}}{d x^{2}}\left(c_{1} \cosh k x+c_{2} \sinh k x\right)=\)

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ \frac{d^{2} y}{d x^{2}}=\sqrt{1+\left(\frac{d y}{d x}\right)^{2}} $$

In Problems 5 and 6, compute \(y^{\prime}\) and \(y^{\prime \prime}\) and then combine these derivatives with \(y\) as a linear second-order differential equation that is free of the symbols \(c_{1}\) and \(c_{2}\) and has the form \(F\left(y, y^{\prime}, y^{\prime \prime}\right)=0\). The symbols \(c_{1}\) and \(c_{2}\) represent constants. $$ y=c_{1} e^{x}+c_{2} x e^{x} $$

\(y=1 /\left(x^{2}+c\right)\) is a one-parameter family of solutions of the first-order DE \(y^{\prime}+2 x y^{2}=0\). Find a 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. $$ y\left(\frac{1}{2}\right)=-4 $$

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with (6). $$ \frac{d^{2} R}{d t^{2}}=-\frac{k}{R^{2}} $$

In Problems 5 and 6, compute \(y^{\prime}\) and \(y^{\prime \prime}\) and then combine these derivatives with \(y\) as a linear second-order differential equation that is free of the symbols \(c_{1}\) and \(c_{2}\) and has the form \(F\left(y, y^{\prime}, y^{\prime \prime}\right)=0\). The symbols \(c_{1}\) and \(c_{2}\) represent constants. $$ y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x $$

Suppose a student carrying a flu virus returns to an isolated college campus of 1000 students. Determine a differential equation governing the number of students \(x(t)\) who have contracted the flu if the rate at which the disease spreads is proportional to the number of interactions between the number of students with the flu and the number of students who have not yet been exposed to it.

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ (\sin \theta) y^{\prime \prime \prime}-(\cos \theta) y^{\prime}=2 $$

In Problems \(7-10, x=c_{1} \cos t+c_{2} \sin t\) is a two-parameter family of solutions of the second-order DE \(x^{\prime \prime}+x=0\). Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ x(0)=-1, \quad x^{\prime}(0)=8 $$

At a time \(t=0\), a technological innovation is introduced into a community with a fixed population of \(n\) people. Determine a differential equation governing the number of people \(x(t)\) who have adopted the innovation at time \(t\) if it is assumed that the rate at which the innovation spreads through the community is jointly proportional to the number of people who have adopted it and the number of people who have not adopted it.

State the order of the given ordinary differential equation. Determine whether the equation is linear or nonlinear by matching it with. $$ \ddot{x}-\left(1-\frac{1}{3} \dot{x}^{2}\right) \dot{x}+x=0 $$

In Problems \(7-10, x=c_{1} \cos t+c_{2} \sin t\) is a two-parameter family of solutions of the second-order DE \(x^{\prime \prime}+x=0\). Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ x(\pi / 2)=0, \quad x^{\prime}(\pi / 2)=1 $$

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ y^{\prime}=2 $$

Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Pure water is pumped into the tank at a rate of \(3 \mathrm{gal} / \mathrm{min}\), and when the solution is well stirred, it is pumped out at the same rate. Determine a differential equation for the amount \(A(t)\) of salt in the tank at time \(t\). What is \(A(0)\) ?

In Problems \(7-10, x=c_{1} \cos t+c_{2} \sin t\) is a two-parameter family of solutions of the second-order DE \(x^{\prime \prime}+x=0\). Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ x(\pi / 6)=\frac{1}{2}, \quad x^{\prime}(\pi / 6)=0 $$

Determine whether the given first-order differential equation is linear in the indicated dependent variable by matching it with the first differential equation given in (7). $$ \left(y^{2}-1\right) d x+x d y=0 ; \text { in } y ; \text { in } x $$

Suppose that a large mixing tank initially holds 300 gallons of water in which 50 pounds of salt has been dissolved. Another brine solution is pumped into the tank at a rate of \(3 \mathrm{gal} / \mathrm{min}\), and when the solution is well stirred, it is pumped out at a slower rate of \(2 \mathrm{gal} / \mathrm{min}\). If the concentration of the solution entering is \(2 \mathrm{lb} / \mathrm{gal}\), determine a differential equation for the amount \(A(t)\) of salt in the tank at time \(t\).

In Problems \(7-10, x=c_{1} \cos t+c_{2} \sin t\) is a two-parameter family of solutions of the second-order DE \(x^{\prime \prime}+x=0\). Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ x(\pi / 4)=\sqrt{2}, \quad x^{\prime}(\pi / 4)=2 \sqrt{2} $$

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ x y^{\prime}=y $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ 2 y^{\prime}+y=0 ; \quad y=e^{-x / 2} $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(0)=1, \quad y^{\prime}(0)=2 $$

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ y^{\prime \prime}+9 y=18 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ \frac{d y}{d t}+20 y=24 ; \quad y=\frac{6}{5}-\frac{6}{5} e^{-20 t} $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(1)=0, \quad y^{\prime}(1)=e $$

In Problems 7-12, match each of the given differential equations with one or more of these solutions: (a) \(y=0\), (b) \(y=2\) (c) \(y=2 x\) (d) \(y=2 x^{2}\). $$ x y^{\prime \prime}-y^{\prime}=0 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime \prime}-6 y^{\prime}+13 y=0 ; \quad y=e^{3 x} \cos 2 x $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(-1)=5, \quad y^{\prime}(-1)=-5 $$

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. $$ y^{\prime \prime}=y^{\prime} $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime \prime}+y=\tan x ; \quad y=-(\cos x) \ln (\sec x+\tan x) $$

y=c_{1} e^{x}+c_{2} e^{-x}\( is a two-parameter family of solutions of the second-order DE \)y^{\prime \prime}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions. $$ y(0)=0, \quad y^{\prime}(0)=0 $$

In Problems 13 and 14, determine by inspection at least one solution of the given differential equation. $$ y^{\prime}=y(y-3) $$

Verify that the indicated function \(y=\phi(x)\) is an explicit solution of the given first-order differential equation. Proceed as in Example 5, by considering \(\phi\) simply as a function, give its domain. Then by considering \(\phi\) as a solution of the differential equation, give at least one interval \(I\) of definition. $$ (y-x) y^{\prime}=y-x+8 ; \quad y=x+4 \sqrt{x}+2 $$

Determine by inspection at least two solutions of the given first-order IVP. $$ y^{\prime}=3 y^{2 / 3}, \quad y(0)=0 $$

$$ y^{\prime}=3 y^{2 / 3}, \quad y(0)=0 $$

In Problems 15 and 16, interpret each statement as a differential equation. On the graph of \(y=\phi(x)\), the slope of the tangent line at a point \(P(x, y)\) is the square of the distance from \(P(x, y)\) to the origin.

A series circuit contains a resistor and a capacitor as shown. Determine a differential equation for the charge \(q(t)\) on the capacitor if the resistance is \(R\), the capacitance is \(C\), and the impressed voltage is \(E(t)\).

Determine by inspection at least two solutions of the given first-order IVP. $$ x y^{\prime}=2 y, \quad y(0)=0 $$

$$ x y^{\prime}=2 y, \quad y(0)=0 $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime}=25+y^{2} ; \quad y=5 \tan 5 x $$

For high-speed motion through the air-such as the skydiver shown falling before the parachute is opened air resistance is closer to a power of the instantaneous velocity \(v(t)\). Determine a differential equation for the velocity \(v(t)\) of a falling body of mass \(m\) if air resistance is proportional to the square of the instantaneous velocity.

Determine a region of the \(x y\)-plane for which the given differential equation would have a unique solution whose graph passes through a point \(\left(x_{0}, y_{0}\right)\) in the region. $$ \frac{d y}{d x}=y^{2 / 3} $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ y^{\prime}=2 x y^{2} ; \quad y=1 /\left(4-x^{2}\right) $$