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Ch. 9 - Differential Equations
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 9 - Differential EquationsProblem 9.2.36a
Chapter 9, Problem 9.2.36a

33–36. {Use of Tech} Computing Euler approximations Use a calculator or computer program to carry out the following steps.
a. Approximate the value of y(T) using Euler’s method with the given time step on the interval [0,T].


y′(t) = t/y, y(0) = 4; Δt = 0.1, T = 2; y(t) = √(t² + 16)

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Identify the differential equation and initial condition: \(y'(t) = \frac{t}{y}\) with \(y(0) = 4\).
Set the step size \(\Delta t = 0.1\) and the interval from \(t=0\) to \(T=2\). Determine the number of steps \(n = \frac{T - 0}{\Delta t} = \frac{2}{0.1} = 20\).
Use Euler's method formula to approximate \(y\) at each step: \(y_{k+1} = y_k + \Delta t \cdot f(t_k, y_k)\), where \(f(t, y) = \frac{t}{y}\).
Start with the initial values \(t_0 = 0\) and \(y_0 = 4\). For each step \(k\) from 0 to 19, compute \(y_{k+1}\) using the formula and update \(t_{k+1} = t_k + \Delta t\).
After completing all steps, the value \(y_{20}\) will be the Euler approximation of \(y(2)\). Compare this approximation to the exact solution \(y(t) = \sqrt{t^2 + 16}\) if desired.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Euler's Method

Euler's method is a numerical technique to approximate solutions of first-order differential equations. It uses a stepwise approach, updating the solution by moving along the slope given by the derivative at each step. This method is especially useful when an exact solution is difficult to find.
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Euler's Method

Initial Value Problems (IVP)

An initial value problem specifies the value of the solution at a starting point, allowing the differential equation to be solved uniquely. Here, y(0) = 4 sets the initial condition, which is essential for applying Euler's method to approximate y(t) over the interval.
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Initial Value Problems

Differential Equation and Exact Solution

The differential equation y′(t) = t/y relates the rate of change of y to t and y itself. The exact solution y(t) = √(t² + 16) provides a benchmark to compare the accuracy of Euler's approximation, helping to understand the method's precision and error.
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Solutions to Basic Differential Equations
Related Practice
Textbook Question

43–44. Motion in a gravitational field: An object is fired vertically upward with initial velocity v(0)=v₀ from initial position s(0)=s₀.

a. For the following values of v₀ and s₀, find the position and velocity functions for all times at which the object is above the ground (s = 0).

v₀ = 49 m/s, s₀ = 60 m

Textbook Question

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.

a. Find the equilibrium solutions. 


y′(t) = y(y - 3)(y + 2)

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Textbook Question

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


a. Find the solutions that are constant, for all t ≥ 0 (the equilibrium solutions).


y'(t) = (y−2)(y+1)

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Textbook Question

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


a. Find the approximations to y(0.2) and y(0.4) using Euler’s method with time steps of Δt = 0.2, 0.1, 0.05, and 0.025.


y′(t) = y/2, y(0) = 2; y(t) = 2eᵗᐟ²

Textbook Question

{Use of Tech} Endowment model An endowment is an investment account in which the balance ideally remains constant and withdrawals are made on the interest earned by the account. Such an account may be modeled by the initial value problem B′(t)=rB−m, for t≥0, with B(0)=B0. The constant r>0 reflects the annual interest rate, m>0 is the annual rate of withdrawal, B0 is the initial balance in the account, and t is measured in years.


a. Solve the initial value problem with r=0.05, m=\(1000/year, and B0=\)15,000 Does the balance in the account increase or decrease?

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Textbook Question

23–26. Stirred tank reactions For each of the following stirred tank reactions, carry out the following analysis.

a. Write an initial value problem for the mass of the substance.


A one-million-liter pond is contaminated by a chemical pollutant with a concentration of 20 g/L. The source of the pollutant is removed, and pure water is allowed to flow into the pond at a rate of 1200 L/hr. Assuming the pond is thoroughly mixed and drained at a rate of 1200 L/hr, how long does it take to reduce the concentration of the solution in the pond to 10% of the initial value?