Chapter 30

A quantity grows with time such that its rate of growth \(d y / d t\) is proportional to the present amount \(y .\) Use this statement to derive the equation for exponential growth, \(y=a e^{n t}\)

Find the general solution to each differential equation. $$(x-y) d x-2 x d y=0$$

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$\frac{d y}{d x}+3 x y=5$$

A biomedical company finds that a certain bacterium used for crop insect control will grow exponentially at the rate of \(12.0 \%\) per hour. Starting with 1000 bacteria, how many will the company have after \(10.0 \mathrm{h} ?\)

The slope of a certain curve at any point is equal to the reciprocal of the ordinate at the point. Write the equation of the curve if it passes through the point (1,3).

Find the general solution of each differential equation. Try some by calculator. $$x d y=(4-y) d x$$

Find the general solution to each differential equation. $$(3 y-x) d x=(x+y) d y$$

Solve each differential equation by calculator. $$y^{\prime}=2 y / x$$

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$y^{\prime \prime}+3 y^{\prime}=5 x$$

If the U.S. energy consumption in 2000 was 158 million barrels (bbl) per day oil equivalent and is growing exponentially at a rate of \(6.9 \%\) per year, estimate the daily oil consumption in the year \(2020 .\)

Find the general solution of each differential equation. Try some by calculator. $$x \frac{d y}{d x}=3-y$$

Find the equation of a curve whose slope at any point is equal to the abscissa of that point divided by the ordinate and which passes through the point (3,4).

Find the general solution to each differential equation. $$\left(x^{2}-x y\right) y^{\prime}+y^{2}=0$$

Solve each differential equation by calculator. $$y^{\prime}=x^{2} / y^{3}$$

A quantity decreases with time such that its rate of decrease \(d y / d t\) is proportional to the present amount \(y .\) Use this statement to derive the equation for exponential decay, \(y=a e^{-n t}\)

Find the general solution of each differential equation. Try some by calculator. $$y+x y^{\prime}=9$$

Find the equation of a curve that passes through (1,1) and whose slope at any point is equal to the product of the ordinate and abscissa.

Find the general solution to each differential equation. $$\frac{d y}{d x}+x y=2 x$$

Solve each differential equation by calculator. $$y^{\prime}=x / 4 y \quad y(5)=2$$

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$\frac{\partial^{2} y}{\partial x^{2}}+4 y=7$$

An iron ingot is \(1850^{\circ} \mathrm{F}\) above room temperature. If it cools exponentially at \(3.50 \%\) per minute, find its temperature (above room temperature) after \(2.50 \mathrm{h}\)

Find the general solution of each differential equation. Try some by calculator. $$2 x y^{\prime}=x-2 y$$

A curve passes through the point (2,3) and has a slope equal to the sum of the abscissa and ordinate at each point. Find its equation.

Find the general solution to each differential equation. Try some by calculator. $$y^{\prime}=\frac{x^{2}}{y^{3}}$$

Solve each differential equation by calculator. $$y^{\prime}=x^{2} / y^{2} \quad y(0)=1$$

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$3\left(y^{\prime \prime}\right)^{4}-5 y^{\prime}=3 y$$

Find the general solution of each differential equation. Try some by calculator. $$x \frac{d y}{d x}=2 x-y$$

A certain pulley in a tape drive is rotating at 2550 rev/min. After the power is shut off, its speed decreases exponentially at a rate of \(12.5 \%\) per second. Find the pulley's speed after \(5.00 \mathrm{s}\)

Find the general solution to each differential equation. $$2 x^{3} y^{\prime}+y^{3}-x^{2} y=0$$

Solve each differential equation by calculator. $$y^{\prime}=3 x / y^{2} \quad y(4)=1$$

Give the order and degree of each equation, and state whether it is an ordinary or partial differential equation. $$4 \frac{d y}{d x}-3\left(\frac{d^{2} y}{d x^{2}}\right)^{3}=x^{2} y$$

Find the general solution of each differential equation. Try some by calculator. $$x d y=\left(3 x^{2}+y\right) d x$$

A forging, initially at \(0^{\circ} \mathrm{F}\), is placed in a furnace at \(1550^{\circ} \mathrm{F}\), where its temperature rises exponentially at the rate of \(6.50 \%\) per minute. Find its temperature after 25.0 min.

Find the general solution to each differential equation. $$y^{\prime}=\frac{3-x y}{2 x^{2}}$$

Using the given boundary condition, find the particular solution. $$x-y=2 x y^{\prime}, x=1, y=1$$

Solve each differential equation and find the approximate value of \(y\) requested. Start at the given boundary value and use a slope field or Euler's graphical or numerical method, as directed by your instructor. \(y^{\prime}=x\) Start at \((0,1) .\) Find \(y(2)\)

Solve each differential equation. $$\frac{d y}{d x}=7 x$$

Find the general solution of each differential equation. Try some by calculator. $$(x+y) d x+x d y=0$$

If we assume that the compressive strength of concrete increases exponentially with time to an upper limit of 4000 lb/in. \(^{2}\), and that the rate of increase is \(52.5 \%\) per week, find the strength after 2 weeks.

Find the general solution to each differential equation. $$y^{\prime}=x+\frac{2 y}{x}$$

Using the given boundary condition, find the particular solution. $$3 x y^{2} d y=\left(3 y^{3}-x^{3}\right) d x, x=1, y=2$$

Solve each differential equation and find the approximate value of \(y\) requested. Start at the given boundary value and use a slope field or Euler's graphical or numerical method, as directed by your instructor. \(y^{\prime}=y\) Start at \((0,1) .\) Find \(y(3)\).

Solve each differential equation. $$2 y^{\prime}=x^{2}$$

Find the general solution of each differential equation. Try some by calculator. $$3 x^{2}+2 y+2 x y^{\prime}=0$$

Find the general solution to each differential equation. $$x y^{\prime}=2 y-x$$

Using the given boundary condition, find the particular solution. $$\left(x^{3}+y^{3}\right) d x-x y^{2} d y=0, x=1, y=0$$

Solve each differential equation and find the approximate value of \(y\) requested. Start at the given boundary value and use a slope field or Euler's graphical or numerical method, as directed by your instructor. \(y^{\prime}=x-2 y\) Start at \((0,4) .\) Find \(y(3)\).

Solve each differential equation. $$4 x-3 y^{\prime}=5$$

Find the general solution of each differential equation. Try some by calculator. $$\left(1-2 x^{2} y\right) \frac{d y}{d x}=2 x y^{2}$$

Find the general solution to each differential equation. Try some by calculator. $$y^{\prime}=x^{2} e^{-3 y}$$