Chapter 2

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(2 y \sin x \cos x-y+2 y^{2} e^{x^{2}}\right) d x=\left(x-\sin ^{2} x-4 x y e^{x^{2}}\right) d y $$

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}-(1+x) y=x y^{2} $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d N}{d t}+N=N t e^{t+2} $$

A dead body was found within a closed room of a bouse where the temperature was a constant \(70^{\circ} \mathrm{F}\). At the time of discovery, the core temperature of the body was determined to be \(85^{\circ} \mathrm{F}\). One hour later a second measurement showed that the core temperature of the body was \(80^{\circ} \mathrm{F}\). Assume that the time of death corresponds to \(t=0\) and that the core temperature at that time was \(98.6^{\circ} \mathrm{F}\). Determine how many hours elapsed before the body was found.

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ (x+1) \frac{d y}{d x}+(x+2) y=2 x e^{-x} $$

Solve the given differential equation by using an appropriate substitution. $$ t^{2} \frac{d y}{d t}+y^{2}=t y $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(4 t^{3} y-15 t^{2}-y\right) d t+\left(t^{4}+3 y^{2}-t\right) d y=0 $$

\(\frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8}\)

Consider the autonomous first-order differential equation \(d y / d x=y-y^{3}\) and the initial condition \(y(0)=y_{0} .\) By hand, sketch the graph of a typical solution \(y(x)\) when \(y_{0}\) has the given values. (a) \(y_{0}>1\) (b) \(0

(a) Without solving, explain why the initial-value problem $$ \frac{d y}{d x}=\sqrt{y}, \quad y\left(x_{0}\right)=y_{0}, $$ has no solution for \(y_{0}<0\). (b) Solve the initial-value problem in part (a) for \(y_{0}>0\) and find the largest interval \(I\) on which the solution is defined.

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ t^{2} \frac{d y}{d t}+y^{2}=t y $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=\frac{x y+3 x-y-3}{x y-2 x+4 y-8} $$

The differential equation $$ \frac{d y}{d x}=\frac{-x+\sqrt{x^{2}+y^{2}}}{y} $$ describes the shape of a plane curve \(C\) that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See Problem 29 in Exercises \(1.3\). There are several ways of solving this \(\mathrm{DE}\). (a) Verify that the differential equation is homogeneous (see Section 2.5). Show that the substitution \(y=u x\) yields $$ \frac{u d u}{\sqrt{1+u^{2}}\left(1-\sqrt{\left.1+u^{2}\right)}\right.}=\frac{d x}{x}. $$ Use a CAS (or another judicious substitution) to integrate the left-hand side of the equation. Show that the curve \(C\) must be a parabola with focus at the origin and is symmetric with respect to the \(x\) -axis. (b) Show that the first differential equation can also be solved by means of the substitution \(u=x^{2}+y^{2}\).

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ (x+2)^{2} \frac{d y}{d x}=5-8 y-4 x y $$

Solve the given differential equation by using an appropriate substitution. $$ 3\left(1+t^{2}\right) \frac{d y}{d t}=2 t y\left(y^{3}-1\right) $$

Determine whether the given differential equation is exact. If it is exact, solve it. $$ \left(\frac{1}{t}+\frac{1}{t^{2}}-\frac{y}{t^{2}+y^{2}}\right) d t+\left(y e^{y}+\frac{1}{t^{2}+y^{2}}\right) d y=0 $$

\(\frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3}\)

Consider the autonomous first-order differential equation \(d y / d x=y^{2}-y^{4}\) and the initial condition \(y(0)=y_{0} .\) By hand, sketch the graph of a typical solution \(y(x)\) when \(y_{0}\) has the given values. (a) \(y_{0}>1\) (b) \(0

(a) Find an implicit solution of the initial-value problem $$ \frac{d y}{d x}=\frac{y^{2}-x^{2}}{x y}, \quad y(1)=-\sqrt{2} $$ (b) Find an explicit solution of the problem in part (a) and give the largest interval \(I\) over which the solution is defined. A graphing utility may be helpful here.

Solar Collector The differential equation $$ \frac{d y}{d x}=\frac{-x+\sqrt{x^{2}+y^{2}}}{y} $$ describes the shape of a plane curve \(C\) that will reflect all incoming light beams to the same point and could be a model for the mirror of a reflecting telescope, a satellite antenna, or a solar collector. See Problem 29 in Exercises 1.3. There are several ways of solving this DE. (a) Verify that the differential equation is homogeneous (see Section 2.5). Show that the substitution \(y=u x\) yields $$ \frac{u d u}{\sqrt{1+u^{2}}\left(1-\sqrt{1+u^{2}}\right)}=\frac{d x}{x} $$ Use a CAS (or another judicious substitution) to integrate the left-hand side of the equation. Show that the curve \(C\) must be a parabola with focus at the origin and is symmetric with respect to the \(x\)-axis. (b) Show that the first differential equation can also be solved by means of the substitution \(u=x^{2}+y^{2}\).

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ 3\left(1+t^{2}\right) \frac{d y}{d t}=2 t y\left(y^{3}-1\right) $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=\frac{x y+2 y-x-2}{x y-3 y+x-3} $$

A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of \(4 \mathrm{~L} / \mathrm{min} ;\) the well-mixed solution is pumped out at the same rate. Find the number \(A(t)\) of grams of salt in the tank at time \(t\).

(a) A simple model for the shape of a tsunami is given by $$ \frac{d W}{d x}=W \sqrt{4-2 W}, $$ where \(W(x)>0\) is the height of the wave expressed as a function of its position relative to a point offshore. By inspection, find all constant solutions of the \(\mathrm{DE}\). (b) Solve the differential equation in part (a). A CAS may be useful for integration. (c) Use a graphing utility to obtain the graphs of all solutions that satisfy the initial condition \(W(0)=2\).

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \frac{d r}{d \theta}+r \sec \theta=\cos \theta $$

Solve the given initial-value problem. $$ x^{2} \frac{d y}{d x}-2 x y=3 y^{4}, \quad y(1)=\frac{1}{2} $$

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

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y^{2}-3 y $$

Tsunami (a) A simple model for the shape of a tsunami is given by $$ \frac{d W}{d x}=W \sqrt{4-2 W} $$ where \(W(x)>0\) is the height of the wave expressed as a function of its position relative to a point offshore. By inspection, find all constant solutions of the DE. (b) Solve the differential equation in part (a). A CAS may be useful for integration. (c) Use a graphing utility to obtain the graphs of all solutions that satisfy the initial condition \(W(0)=2\).

Each \(D E\) in Problems \(15-22\) is a Bernoulli equation. In Problems 15-20, solve the given differential equation by using an appropriate substitution. $$ y^{1 / 2} \frac{d y}{d x}+y^{3 / 2}=1, \quad y(0)=4 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=x \sqrt{1-y^{2}} $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \frac{d r}{d \theta}+r \sec \theta=\cos \theta $$

Solve the given initial-value problem. $$ y^{1 / 2} \frac{d y}{d x}+y^{3 / 2}=1, \quad y(0)=4 $$

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

\(\left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2}\)

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=y^{2}-y^{3} $$

Use Euler's method with step size \(h=0.1\) to approximate \(y(1.2)\) where \(y(x)\) is a solution of the initial-value problem \(y^{\prime}=1+x \sqrt{y}, y(1)=9\).

In Problems 1-22, solve the given differential equation by separation of variables. $$ \left(e^{x}+e^{-x}\right) \frac{d y}{d x}=y^{2} $$

A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of \(5 \mathrm{gal} / \mathrm{min}\). The well-mixed solution is pumped out at the same rate. Find the number \(A(t)\) of pounds of salt in the tank at time \(t\).

Solve the given initial-value problem. $$ (4 y+2 t-5) d t+(6 y+4 t-1) d y=0, \quad y(-1)=2 $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(\frac{d x}{d t}=4\left(x^{2}+1\right), \quad x(\pi / 4)=1\)

Find the critical points and phase portrait of the given autonomous first- order differential equation. Classify each critical point as asymptotically stable, unstable, or semi-stable. By hand, sketch typical solution curves in the regions in the \(x y\) -plane determined by the graphs of the equilibrium solutions. $$ \frac{d y}{d x}=(y-2)^{4} $$

In March 1976, the world population reached 4 billion. A popular news magazine predicted that with an average yearly growth rate of \(1.8 \%\), the world population would be 8 billion in 45 years. How does this value compare with that predicted by the model that says the rate of increase is proportional to the population at any time \(t\) ?

In Problems 23-28, find an implicit and an explicit solution of the given initial-value problem. $$ \frac{d x}{d t}=4\left(x^{2}+1\right), \quad x(\pi / 4)=1 $$

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ x \frac{d y}{d x}+(3 x+1) y=e^{-3 x} $$

(a) In Examples 3 and 4 of Section 2.1, we saw that any solution \(P(t)\) of \((4)\) possesses the asymptotic behavior \(P(t) \rightarrow a / b\) as \(t \rightarrow \infty\) for \(P_{0}>a / b\) and for \(0

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution. $$ \left(x^{2}-1\right) \frac{d y}{d x}+2 y=(x+1)^{2} $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{1-x-y}{x+y} $$

Solve the given initial-value problem. $$ \left(\frac{3 y^{2}-t^{2}}{y^{5}}\right) \frac{d y}{d t}+\frac{t}{2 y^{4}}=0, \quad y(1)=1 $$

In Problems, find an implicit and an explicit solution of the given initial- value problem. \(\frac{d y}{d x}=\frac{y^{2}-1}{x^{2}-1}, \quad y(2)=2\)