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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.RE.1a
Chapter 9, Problem 9.RE.1a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The differential equation y′+2y=t is first-order, linear, and separable.

Verified step by step guidance
1
Identify the order of the differential equation \(y' + 2y = t\). Since the highest derivative present is \(y'\), this is a first-order differential equation.
Check if the equation is linear. A first-order linear differential equation can be written in the form \(y' + p(t)y = q(t)\), where \(p(t)\) and \(q(t)\) are functions of \(t\). Here, \(p(t) = 2\) and \(q(t) = t\), so the equation is linear.
Determine if the equation is separable. A separable differential equation can be written as \(\frac{dy}{dt} = g(t)h(y)\), allowing the variables \(y\) and \(t\) to be separated on opposite sides of the equation.
Rewrite the given equation as \(y' = t - 2y\). Notice that the right side is \(t - 2y\), which is a sum of a function of \(t\) and a function of \(y\), not a product of separate functions of \(t\) and \(y\).
Since the right side cannot be expressed as a product \(g(t)h(y)\), the equation is not separable. Therefore, the equation is first-order and linear, but not separable.

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Key Concepts

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

First-Order Differential Equations

A first-order differential equation involves only the first derivative of the unknown function and no higher derivatives. It can be written in the form dy/dt = f(t, y), where the highest derivative is of order one. Recognizing the order helps classify and solve the equation.
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Classifying Differential Equations

Linear Differential Equations

A differential equation is linear if the unknown function and its derivatives appear to the first power and are not multiplied together. It can be expressed as y' + p(t)y = q(t), where p(t) and q(t) are functions of the independent variable. Linearity allows the use of specific solution methods like integrating factors.
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Classifying Differential Equations

Separable Differential Equations

A separable differential equation can be written so that all terms involving y are on one side and all terms involving t are on the other, allowing integration of each side separately. It has the form dy/dt = g(t)h(y). If the equation cannot be rearranged this way, it is not separable.
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Solving Separable Differential Equations
Related Practice
Textbook Question

22–25. Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable.

y′(t) = y(3+y)(y-5)

Textbook Question

Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.

b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.

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Textbook Question

2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.

y′(t) = √(y/t)

Textbook Question

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.


a. Show that t₁/₂ = −1/k ln((T₀ − 2A)/(2(T₀ − A))).

Textbook Question

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

d. The direction field for the differential equation y′(t)=t+y(t) is plotted in the ty-plane.

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Textbook Question

Stirred tank reaction A 100-L tank is filled with pure water when an inflow pipe is opened and a sugar solution with a concentration of 20 gm/L flows into the tank at a rate of 0.5 L/min. The solution is thoroughly mixed and flows out of the tank at a rate of 0.5 L/min.


c. At what time does the mass of sugar reach 95% of its steady-state level?

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