{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R\u003e0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).c. Find the solution of this separable equation assuming v(0)=0 and 0\u003cv\u003cg/b .

Ch. 9 - Differential Equations

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 9 - Differential Equations

Problema 9.3.48c

Capitolo 9, Problema 9.3.48c

{Use of Tech} Free fall Using th e background given in Exercise 47, assume the resistance is given by f(v)=−Rv, for t≥0, where R>0 is a drag coefficient (an assumption often made for a heavy medium such as water or oil).

c. Find the solution of this separable equation assuming v(0)=0 and 0<v<g/b.

Guida verificata passo dopo passo

Start with the given differential equation for velocity under resistance: \[\frac{dv}{dt} = g - bv - Rv,\] where $g$ is the acceleration due to gravity, $b$ is a constant, and $R > 0$ is the drag coefficient. Combine the terms involving $v$ to rewrite the equation as \[\frac{dv}{dt} = g - (b + R)v.\]

Recognize that this is a first-order linear ordinary differential equation in $v(t)$, which can be rearranged as \[\frac{dv}{dt} + (b + R)v = g.\]

To solve this linear ODE, find the integrating factor $\mu(t)$, which is given by \[\mu(t) = e^{\int (b + R) dt} = e^{(b + R)t}.\]

Multiply both sides of the differential equation by the integrating factor to get \[e^{(b + R)t} \frac{dv}{dt} + (b + R) e^{(b + R)t} v = g e^{(b + R)t},\] which simplifies to \[\frac{d}{dt} \left( e^{(b + R)t} v \right) = g e^{(b + R)t}.\]

Integrate both sides with respect to $t$ to find \[e^{(b + R)t} v = \int g e^{(b + R)t} dt + C,\] where $C$ is the constant of integration. Then solve for $v(t)$ by dividing both sides by $e^{(b + R)t}$. Use the initial condition $v(0) = 0$ to determine $C$.

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Separable Differential Equations

A separable differential equation can be written as a product of a function of the dependent variable and a function of the independent variable. This allows the variables to be separated on opposite sides of the equation and integrated independently, simplifying the solution process.

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Solving Separable Differential Equations

Modeling Drag Force as Proportional to Velocity

In fluid resistance problems, drag force is often modeled as proportional to velocity, expressed as f(v) = -Rv, where R is the drag coefficient. This linear resistance opposes motion and affects the acceleration and velocity of the falling object.

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Exponential Growth & Decay

Initial Conditions in Differential Equations

Initial conditions specify the value of the solution at a particular point, such as v(0) = 0. They are essential for determining the unique solution to a differential equation from the family of possible solutions obtained after integration.

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Solutions to Basic Differential Equations

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38–43. Equilibrium solutions A differential equation of the form y′(t)=f(y) is said to be autonomous (the function f depends only on y). The constant function y=y0 is an equilibrium solution of the equation provided f(y0)=0 (because then y'(t)=0 and the solution remains constant for all t). Note that equilibrium solutions correspond to horizontal lines in the direction field. Note also that for autonomous equations, the direction field is independent of t. Carry out the following analysis on the given equations.

c. Sketch the solution curve that corresponds to the initial condition y0=1.

y′(t) = 2y + 4

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29–32. {Use of Tech} Errors in Euler’s method Consider the following initial value problems.

c. Which time step results in the more accurate approximation? Explain your observations.

y′(t) = 4−y, y(0) = 3; y(t) = 4−e⁻ᵗ

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27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.

c. Find the equilibrium points for the system.

x′(t) = −3x + xy, y′(t) = 2y − xy

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Cooling time Suppose an object with an initial temperature of T₀ > 0 is put in surroundings with an ambient temperature of A, where A < T₀/2. Let t₁/₂ be the time required for the object to cool to T₀/2.

c. Why is the condition A < T₀/2 needed?

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33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.

d. Compare the errors in the approximations to y(T).

y′(t) = 6 - 2y, y(0) = -1; Δt = 0.2, T = 3; y(t) = 3 - 4e⁻²ᵗ

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17–20. Increasing and decreasing solutions Consider the following differential equations. A detailed direction field is not needed.

d. Sketch the direction field and verify that it is consistent with parts (a)–(c).

y'(t) = (y−2)(y+1)

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