Chapter 1

(a) Give the domain of the function \(y=x^{2 / 3}\). (b) Give the largest interval \(I\) of definition over which \(y=x^{2 / 3}\) is a solution of the differential equation \(3 x y^{\prime}-2 y=0\).

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. $$ 2 y^{\prime}=y^{3} \cos x ; \quad y=(1-\sin x)^{-1 / 2} $$

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. $$ \frac{d y}{d x}=\sqrt{x y} $$

Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ 2 y^{\prime}=y^{3} \cos x ; \quad y=(1-\sin x)^{-1 / 2} $$

(a) Verify that the one-parameter family \(y^{2}-2 y=x^{2}-\) \(x+c\) is an implicit solution of the differential equation \((2 y-2) y^{\prime}=2 x-1\). (b) Find a member of the one-parameter family in part (a) that satisfies the initial condition \(y(0)=1\). (c) Use your result in part (b) to find an explicit function \(y=\phi(x)\) that satisfies \(y(0)=1\). Give the domain of \(\phi\). Is \(y=\phi(x)\) a solution of the initial-value problem? If so, give its interval \(I\) of definition; if not, explain.

In Problems 19 and 20, verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of the definition of each solution \(\phi\). $$ \frac{d X}{d t}=(X-1)(1-2 X) ; \quad \ln \left(\frac{2 X-1}{X-1}\right)=t $$

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. $$ x \frac{d y}{d x}=y $$

Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi\). $$ \frac{d X}{d t}=(X-1)(1-2 X) ; \quad \ln \left(\frac{2 X-1}{X-1}\right)=t $$

Given that \(y=-\frac{2}{x}+x\) is a solution of the \(\mathrm{DE} x y^{\prime}+y=2 x\). Find \(x_{0}\) and the largest interval \(I\) for which \(y(x)\) is a solution of the IVP $$ x y^{\prime}+y=2 x, \quad y\left(x_{0}\right)=1 . $$

In Problems 19 and 20, verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of the definition of each solution \(\phi\). $$ 2 x y d x+\left(x^{2}-y\right) d y=0 ; \quad-2 x^{2} y+y^{2}=1 $$

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. $$ \frac{d y}{d x}-y=x $$

Verify that the indicated expression is an implicit solution of the given first-order differential equation. Find at least one explicit solution \(y=\phi(x)\) in each case. Use a graphing utility to obtain the graph of an explicit solution. Give an interval \(I\) of definition of each solution \(\phi\). $$ 2 x y d x+\left(x^{2}-y\right) d y=0 ; \quad-2 x^{2} y+y^{2}=1 $$

Suppose that \(y(x)\) denotes a solution of the initial-value problem \(y^{\prime}=x^{2}+y^{2}, y(1)=-1\) and that \(y(x)\) possesses at least a second derivative at \(x=1\). In some neighborhood of \(x=1\), use the DE to determine whether \(y(x)\) is increasing or decreasing, and whether the graph \(y(x)\) is concave up or concave down.

Verify that the indicated family of functions is a solution to the given differential equation. Assume an appropriate interval \(I\) of definition for each solution.$$ \frac{d P}{d t}=P(1-P) ; \quad P=\frac{c_{1} e^{t}}{1+c_{1} e^{t}} $$

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. $$ \left(4-y^{2}\right) y^{\prime}=x^{2} $$

A differential equation may possess more than one family of solutions. (a) Plot different members of the families \(y=\phi_{1}(x)=\) \(x^{2}+c_{1}\) and \(y=\phi_{2}(x)=-x^{2}+c_{2}\) (b) Verify that \(y=\phi_{1}(x)\) and \(y=\phi_{2}(x)\) are two solutions of the nonlinear first-order differential equation \(\left(y^{\prime}\right)^{2}=4 x^{2}\). (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).

Verify that the indicated family of functions is a solution to the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ \frac{d y}{d x}+2 x y=1 ; \quad y=e^{-x^{2}} \int_{0}^{x} e^{t^{2}} d t+c_{1} e^{-x^{2}} $$

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. $$ \left(1+y^{3}\right) y^{\prime}=x^{2} $$

Verify that the indicated family of functions is a solution to the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ \frac{d^{2} y}{d x^{2}}-4 \frac{d y}{d x}+4 y=0 ; \quad y=c_{1} e^{2 x}+c_{2} x e^{2 x} $$

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. $$ \left(x^{2}+y^{2}\right) y^{\prime}=y^{2} $$

In Problems 23-26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition \(I\) for each solution. $$ y^{\prime \prime}+y=2 \cos x-2 \sin x ; \quad y=x \sin x+x \cos x $$

Verify that the indicated family of functions is a solution to the given differential equation. Assume an appropriate interval \(I\) of definition for each solution. $$ \begin{aligned} &x^{3} \frac{d^{3} y}{d x^{3}}+2 x^{2} \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}+y=12 x^{2} \\ &y=c_{1} x^{-1}+c_{2} x+c_{3} x \ln x+4 x^{2} \end{aligned} $$

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. $$ (y-x) y^{\prime}=y+x $$

Verify that the piecewise-defined function $$ y=\left\\{\begin{array}{ll} -x^{2}, & x<0 \\ x^{2}, & x \geq 0 \end{array}\right. $$ is a solution of the differential equation \(x y^{\prime}-2 y=0\) on the interval \((-\infty, \infty)\)

In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose \(M\) denotes the total amount of a subject to be memorized and \(A(t)\) is the amount memorized in time \(t .\) Determine a differential equation for the amount \(A(t)\)

Learning Theory In the theory of learning, the rate at which a subject is memorized is assumed to be proportional to the amount that is left to be memorized. Suppose \(M\) denotes the total amount of a subject to be memorized and \(A(t)\) is the amount memorized in time \(t\). Determine a differential equation for the amount \(A(t)\).

Verify that the piecewise-defined function $$ y= \begin{cases}-x^{2}, & x<0 \\ x^{2}, & x \geq 0\end{cases} $$ is a solution of the differential equation \(x y^{\prime}-2 y=0\) on the interval \((-\infty, \infty)\).

In Problems 23-26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition \(I\) for each solution. $$ x^{2} y^{\prime \prime}+x y^{\prime}+y=0 ; \quad y=\sin (\ln x) $$

In Problems 23-26, verify that the indicated function is an explicit solution of the given differential equation. Give an interval of definition \(I\) for each solution. $$ \begin{aligned} &x^{2} y^{\prime \prime}+x y^{\prime}+y=\sec (\ln x) \\ &y=\cos (\ln x) \ln (\cos (\ln x))+(\ln x) \sin (\ln x) \end{aligned} $$

$$ \text { 27. } y^{\prime}+2 y=0 $$

A drug is infused into a patient's bloodstream at a constant rate of \(r\) grams per second. Simultaneously, the drug is removed at a rate proportional to the amount \(x(t)\) of the drug present at time \(t .\) Determine a differential equation governing the amount \(x(t)\).

Infusion of a Drug A drug is infused into a patient's bloodstream at a constant rate of \(r\) grams per second. Simultaneously, the drug is removed at a rate proportional to the amount \(x(t)\) of the drug present at time \(t\). Determine a differential equation governing the amount \(x(t)\).

Find values of \(m\) so that the function \(y=e^{m x}\) is a solution of the given differential equation. $$ y^{\prime}+2 y=0 $$

In Problems 27-30 verify that the indicated expression is an implicit solution of the given differential equation. $$ x \frac{d y}{d x}+y=\frac{1}{y^{2}} ; \quad x^{3} y^{3}=x^{3}+5 $$

$$ 3 y^{\prime}=4 y $$

Find values of \(m\) so that the function \(y=e^{m x}\) is a solution of the given differential equation. $$ 3 y^{\prime}=4 y $$

$$ y^{\prime \prime}-5 y^{\prime}+6 y=0 $$

(a) Byinspection, find a one-parameter family of solutions of the differential equation \(x y^{\prime}=y .\) Verify that each member of the family is a solution of the initial-value problem \(x y^{\prime}=y, y(0)=0\) (b) Explain part (a) by determining a region \(R\) in the \(x y\) -plane for which the differential equation \(x y^{\prime}=y\) would have a unique solution through a point \(\left(x_{0}, y_{0}\right)\) in \(R\). (c) Verify that the picecwise-defined function $$ y=\left\\{\begin{array}{ll} 0, & x<0 \\ x, & x \geq 0 \end{array}\right. $$ satisfies the condition \(y(0)=0\). Determine whether this function is also a solution of the initial-value problem in part (a).

Find values of \(m\) so that the function \(y=e^{m x}\) is a solution of the given differential equation. $$ y^{\prime \prime}-5 y^{\prime}+6 y=0 $$

(a) Verify that \(y=\tan (x+c)\) is a one-parameter family of solutions of the differential equation \(y^{\prime}=1+y^{2}\). (b) Since \(f(x, y)=1+y^{2}\) and \(\partial f / \partial y=2 y\) are continuous everywhere, the region \(R\) in Theorem \(1.2 .1\) can be taken to be the entire \(x y\) -plane. Use the family of solutions in part (a) to find an explicit solution of the first-order initialvalue problem \(y^{\prime}=1+y^{2}, y(0)=0 .\) Even though \(x_{0}=0\) 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).

Find values of \(m\) so that the function \(y=e^{m x}\) is a solution of the given differential equation. $$ 2 y^{\prime \prime}+9 y^{\prime}-5 y=0 $$

$$ x y^{\prime \prime}+2 y^{\prime}=0 $$

(a) Verify that \(y=-1 /(x+c)\) is a one-parameter family of solutions of the differential equation \(y^{\prime}=y^{2}\). (b) Since \(f(x, y)=y^{2}\) and \(\partial f l \partial y=2 y\) are continuous everywhere, the region \(R\) in Theorem \(1.2 .1\) can be taken to be the entire \(x y\) -plane. Find a solution from the family in part (a) that satisfies \(y(0)=1\). 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.

Use the concept that \(y=c,-\infty

In Problem 31 and 32, verify that the function defined by the definite integral is a particular solution of the given differential equation. In both problems, use the Leibniz formula for the derivative of an integral: \(\frac{d}{d x} \int_{u(x)}^{v(x)} F(x, t) d t=F(x, v(x)) \frac{d v}{d x}-F(x, u(x)) \frac{d u}{d x}+\int_{u(x)}^{v(x)} \frac{\partial}{\partial x} F(x, t) d t\) $$ y^{\prime \prime}+9 y=f(x) ; y(x)=\frac{1}{3} \int_{0}^{x} f(t) \sin 3(x-t) d t $$

Use the concept that \(y=c,-\infty

Use the concept that \(y=c,-\infty

The differential equation \(d P / d t=(k \cos t) P\) where \(k\) is a positive constant, is a model of human population \(P(t)\) of a certain community. Discuss an interpretation for the solution of this equation; in other words, what kind of population do you think the differential equation describes?

(a) Verify that \(3 x^{2}-y^{2}=c\) is a one-parameter family of solutions of the differential equation \(y d y / d x=3 x\). (b) By hand, sketch the graph of the implicit solution \(3 x^{2}-y^{2}=3 .\) Find all explicit solutions \(y=\phi(x)\) 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 \(3 x^{2}-y^{2}=3\), but which of the explicit solutions in part (b) satisfies \(y(-2)=3 ?\)

PopulationModel Thedifferential equation \(d P / d t=(k \cos t) P\) where \(k\) is a positive constant, is a model of human population \(P(t)\) of a certain community. Discuss an interpretation for the solution of this equation; in other words, what kind of population do you think the differential equation describes?