Chapter 2

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(y^{\prime}=x+y\) (a) \(y(-2)=2\) (b) \(y(1)=-3\)

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ \left(y^{2}+y x\right) d x+x^{2} d y=0 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}+2 x y^{2}=0 $$

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0.5 ; y(0.5) $$

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. $$ x^{2} y^{\prime}+x y=1 $$

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

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

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

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{y-x}{y+x} $$

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

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=x y+\sqrt{y}, \quad y(0)=1 ; y(0.5) $$

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. $$ y^{\prime}=2 y+x^{2}+5 $$

Solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{x+3 y}{3 x+y} $$

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

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

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(\frac{d y}{d x}=\frac{1}{y}\) (a) \(y(0)=1\) (b) \(y(-2)=-1\)

Classify each differential equation as separable, exact, linear, homogeneous, or Bernoulli. Some equations may be more than one kind. Do not solve. (a) \(\frac{d y}{d x}=\frac{x-y}{x}\) (b) \(\frac{d y}{d x}=\frac{1}{y-x}\) (c) \((x+1) \frac{d y}{d x}=-y+10\) (d) \(\frac{d y}{d x}=\frac{1}{x(x-y)}\) (e) \(\frac{d y}{d x}=\frac{y^{2}+y}{x^{2}+x}\) (f) \(\frac{d y}{d x}=5 y+y^{2}\) (g) \(y d x=\left(y-x y^{2}\right) d y\) (h) \(x \frac{d y}{d x}=y e^{x y}-x\) (i) \(x y y^{\prime}+y^{2}=2 x\) (j) \(2 x y y^{\prime}+y^{2}=2 x^{2}\) (k) \(y d x+x d y=0\) (l) \(\left(x^{2}+\frac{2 y}{x}\right) d x=\left(3-\ln x^{2}\right) d y\) (m) \(\frac{d y}{d x}=\frac{x}{y}+\frac{y}{x}+1\) (n) \(\frac{y}{x^{2}} \frac{d y}{d x}+e^{2 x^{3}+y^{2}}=0\)

(a) Consider the initial-value problem \(d A / d t=k A, A(0)=A_{0}\), as the model for the decay of a radioactive substance. Show that, in general, the half-life \(T\) of the substance is \(T=-(\ln 2) / k\). (b) Sbow that the solution of the initial-value problemin part (a) can be written \(A(t)=A_{0} 2^{-\int}\). (c) If a radioactive substance has a balf-life \(T\) given in part (a), how long will it tale an initial amount \(A_{0}\) of the substance to decay to \(A_{0}\) ?

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ \frac{d y}{d x}=\frac{x+3 y}{3 x+y} $$

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

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=x y^{2}-\frac{y}{x}, \quad y(1)=1 ; y(1.5) $$

When a vertical beam of light passes through a transparent medium, therate at whichits intensity I decreasesis propartional to \(I(t)\), where trepresents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is \(25 \%\) of the initial intensity \(I_{0}\) of the incident beam. What is the intensity of the beam 15 feet below the surface?

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. $$ x \frac{d y}{d x}-y=x^{2} \sin x $$

Solve the given differential equation by using an appropriate substitution. $$ -y d x+(x+\sqrt{x y}) d y=0 $$

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

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(\frac{d y}{d x}=0.2 x^{2}+y\) (a) \(y(0)=\frac{1}{2}\) (b) \(y(2)=-1\)

In Problems 9-16, solve the given differential equation. $$ \left(y^{2}+1\right) d x=y \sec ^{2} x d y $$

When a vertical beam of light passes throrgh a transparent medium, the rate at which its intensity \(I\) decreasesis proportional \(t o I(t)\), where \(t\) represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is \(25 \%\) of the initial intensity \(I_{0}\) of the incident beam. What is the intensity of the beam 15 feet below the surface?

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

Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use \(h=0.1\) and then use \(h=0.05\). \(-y\) $$ y^{\prime}=y-y^{2}, \quad y(0)=0.5 ; y(0.5) $$

When interest is compounded continuously, the amount of money increases at a rate proportional to the amount \(S\) present at time \(t\), that is, \(d S / d t=r S\), where \(r\) is the annual rate of interest. (a) Find the amount of money accrued at the end of 5 years when \(\$ 5000\) is deposited in a savings account drawing \(5 \frac{3}{4} \%\) annual interest compounded continuously. (b) In how many years will the initial sum deposited bave doubled? (c) Use a calculator to compare the amount obtained in part (a) with the amount \(S=5000\left(1+\frac{1}{4}(0.0575)\right)^{5(4)}\) that is arcrued when interest is compounded quarterly.

Effect For an initial population \(P_{0}\), where \(P_{0}>K\) the logistic population model (3) predicts that population cannot sustain itself over time so it decreases but yet never falls below the carrying capacity \(K\) of the ecosystem. Moreover, 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. $$ x \frac{d y}{d x}+2 y=3 $$

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

\(\frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2}\)

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(\frac{d y}{d x}=x e^{y}\) (a) \(y(0)=-2\) (b) \(y(1)=2.5\)

In Problems 9-16, solve the given differential equation. $$ y(\ln x-\ln y) d x=(x \ln x-x \ln y-y) d y $$

In Problems \(1-10\), solve the given differential equation by using an appropriate substitution. $$ x \frac{d y}{d x}=y+\sqrt{x^{2}-y^{2}}, x>0 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=\left(\frac{2 y+3}{4 x+5}\right)^{2} $$

Use a numerical solver to obtain a numerical solution curve for the given initial-value problem. First use Euler's method and then the RK4 method. Use \(h=0.25\) in each case. Superimpose both solution curves on the same coordinate axes. If possible, use a different color for each curve. Repeat, using \(h=0.1\) and \(h=0.05\). $$ y^{\prime}=2(\cos x) y, \quad y(0)=1 $$

Consider the Lothe -Volterra prailatur-prey model defined by $$ \begin{aligned} &\frac{d x}{d t}=-0.1 x+0.02 x y \\ &\frac{d y}{d t}=0.2 y-0.025 x y \end{aligned} $$ where the populations \(x(t)\) (predatars) and \(y(t)\) (prey) are measured in the thousands. Súfrose \(x(0)=6\) and \(y(0)=6\). Use a numerical solverto graph \(x(t)\) and \(y(t)\). Use the graphs to sqproximate the time \(t>0\) when the two populations are first equal. Use the grapbs to aqproximate the period of each population.

Two chemicals \(A\) and \(B\) are combined to form a chemical \(C\). The rate, or velocity, of the reaction is proportional to the product of the instantaneous amounts of \(A\) and \(B\) not converted to chemical \(C\). Initially there are 40 grams of \(A\) and 50 grams of \(B\), and for each gram of \(B\), 2 grams of \(A\) is used. It is observed that 10 grams of \(C\) is formed in 5 minutes. How much is formed in 20 minutes? What is the limiting amount of \(C\) after a long time? How much of chemicals \(A\) and \(B\) remains after a long time?

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. $$ x \frac{d y}{d x}+4 y=x^{3}-x $$

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

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

Use computer software to obtain a direction field for the given differential equation. By hand, sketch an approximate solution curve passing through each of the given points. \(y^{\prime}=y-\cos \frac{\pi}{2} x\) (a) \(y(2)=2\) (b) \(y(-1)=0\)

In Problems 9-16, solve the given differential equation. $$ (6 x+1) y^{2} \frac{d y}{d x}+3 x^{2}+2 y^{3}=0 $$

Consider the Lotha-Volterra pruatatur-pucy model defined by $$ \begin{aligned} &\frac{d x}{d t}=-0.1 x+0.02 x y \\ &\frac{d y}{d t}=0.2 y-0.025 x y \end{aligned} $$ where the populations \(x(t)\) (predators) and \(y(t)\) (prey) are measured in the thousands. Surpose \(x(0)=6\) and \(y(0)=6\). Use a numerical solver to graph \(x(t)\) and \(y(t)\). Use the graphs to approximate the time \(t>0\) when the two populations are first equal. Use the graphs to approximale the peciod of each population.

$$ \text { In Problems 11-14, solve the given initial-value problem. } $$ $$ x y^{2} \frac{d y}{d x}=y^{3}-x^{3}, y(1)=2 $$

In Problems 1-22, solve the given differential equation by separation of variables. $$ \csc y d x+\sec ^{2} x d y=0 $$