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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 9.5.35b
Chapter 9, Problem 9.5.35b

Properties of stirred tank solutions


b. Verify that M(0) = M₀

Verified step by step guidance
1
Identify the function M(t) that describes the amount of substance in the stirred tank at time t. This function is usually given or derived from the problem context.
Recall that M(0) represents the initial amount of substance in the tank at time t = 0.
Substitute t = 0 into the function M(t) to find M(0). This means replacing every occurrence of t in the expression with 0.
Simplify the resulting expression after substitution to see if it equals M₀, the initial amount given in the problem.
If the simplified expression equals M₀, then you have verified that M(0) = M₀, confirming the initial condition of the solution.

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Key Concepts

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

Initial Condition in Differential Equations

The initial condition specifies the value of the function at the starting point, often time zero. Verifying M(0) = M₀ ensures the solution matches the known initial state, which is crucial for the correctness of models describing physical systems like stirred tanks.
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Mass Balance in Stirred Tank Systems

Mass balance involves accounting for the mass entering, leaving, and accumulating in the tank. Understanding this principle helps formulate the differential equation governing M(t), the mass in the tank, and ensures the model reflects physical reality.
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Solving and Verifying Solutions of Differential Equations

After solving the differential equation for M(t), verifying M(0) = M₀ confirms the solution satisfies the initial condition. This step validates the mathematical model and ensures consistency between the solution and the physical system.
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Verifying Solutions of Differential Equations
Related Practice
Textbook Question

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

b. Using the exact solution (also given), find the error in the approximation to y(T) (only at the right endpoint of the time interval).


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

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Textbook Question

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.

b. Find the lines along which x'(t) = 0. Find the lines along which y'(t) = 0.


x′(t) = 2x − 4xy, y′(t) = −y + 2xy

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Textbook Question

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


b. In what regions are solutions increasing? Decreasing?


y'(t) = (y−1)(1+y)

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. Euler’s method is used to compute exact values of the solution of an initial value problem. 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. The solution of a stirred tank initial value problem always approaches a constant as t→∞

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Textbook Question

Blowup in finite time Consider the initial value problem y'(t) = yⁿ + 1, y(0) = y₀, where n is a positive integer.

b. Solve the initial value problem with n = 2 and y₀ = 1/√2.

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