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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 9.3.6
Chapter 9, Problem 9.3.6

5–16. Solving separable equations Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable.
e⁴ᵗy'(t) = 5

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Identify the given differential equation: \(e^{4t} y'(t) = 5\).
Rewrite the equation to isolate \(y'(t)\): \(y'(t) = \frac{5}{e^{4t}} = 5 e^{-4t}\).
Recognize that this is a separable differential equation where \(y'(t)\) is expressed explicitly in terms of \(t\).
Integrate both sides with respect to \(t\) to find \(y(t)\): \(y(t) = \int 5 e^{-4t} \, dt\).
Perform the integration using the formula \(\int e^{at} dt = \frac{1}{a} e^{at} + C\), and include the constant of integration \(C\) to express the general solution.

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Key Concepts

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

Separable Differential Equations

A separable differential equation can be written so that all terms involving the dependent variable are on one side and all terms involving the independent variable are on the other. This allows integration of both sides separately to find the general solution.
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Solving Separable Differential Equations

Integration of Both Sides

After separating variables, integrate each side with respect to its variable. This step is crucial to find an implicit or explicit form of the solution, often involving an integration constant representing the general solution.
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One-Sided Limits

Expressing the Solution Explicitly

Once integrated, solve the resulting equation for the dependent variable explicitly as a function of the independent variable. This often involves algebraic manipulation to isolate the dependent variable on one side.
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Simplifying Trig Expressions
Related Practice
Textbook Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.


A pot of boiling soup (100°C) is put in a cellar with a temperature of 10°C. After 30 minutes, the soup has cooled to 80°C. When will the temperature of the soup reach 30°C 

Textbook Question

Explain how a stirred tank reaction works.

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Textbook Question

17–32. Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem.

y(t) = sec² t/(2y), y(π/4) = 1

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Textbook Question

Stability of Euler's method Consider the initial value problem y′(t) = −ay, y(0) = 1 where a > 0; it has the exact solution y(t) = e⁻ᵃᵗ, which is a decreasing function.


a. Show that Euler's method applied to this problem with time step h can be written u₀ = 1, uₖ₊₁ = (1 − ah)uₖ for k = 0, 1, 2, ...


b. Show by substitution that uₖ = (1 − ah)ᵏ is a solution of the equations in part (a), for k = 0, 1, 2, ...

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Textbook Question

Explain how to solve a separable differential equation of the form

g(t)y'(y) = h(t)

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Textbook Question

27–30. Newton’s Law of Cooling Solve the differential equation for Newton’s Law of Cooling to find the temperature function in the following cases. Then answer any additional questions.


An iron rod is removed from a blacksmith’s forge at a temperature of 900°C . Assume k=0.02 and the rod cools in a room with a temperature of 30°C When does the temperature of the rod reach 100°C? 

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