Textbook Question
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.
Ch. 9 - Differential Equations
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 9, Problem 9.R.6
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) = √(y/t)
Verified step by step guidance1
Rewrite the differential equation \(y'(t) = \sqrt{\frac{y}{t}}\) in Leibniz notation as \(\frac{dy}{dt} = \sqrt{\frac{y}{t}}\) to clearly see the relationship between \(y\) and \(t\).
Recognize that the equation is separable, meaning you can rearrange terms to isolate \(y\) on one side and \(t\) on the other. Start by expressing \(\frac{dy}{dt} = \frac{\sqrt{y}}{\sqrt{t}}\).
Separate variables by writing \(\frac{dy}{\sqrt{y}} = \frac{dt}{\sqrt{t}}\). This allows you to integrate both sides with respect to their own variables.
Integrate both sides: \(\int \frac{1}{\sqrt{y}} \, dy = \int \frac{1}{\sqrt{t}} \, dt\). Recall that \(\int y^{-1/2} dy = 2\sqrt{y} + C\) and similarly for \(t\).
After integrating, solve the resulting equation for \(y\) to express the general solution explicitly in terms of \(t\) and an arbitrary constant of integration.
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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 as a product of a function of t and a function of y, allowing the variables to be separated on opposite sides of the equation. This enables integration with respect to each variable independently to find the general solution.
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Solving Separable Differential Equations
Integration Techniques
Solving separable equations requires integrating both sides after separation. Familiarity with basic integration rules, including power functions and substitution, is essential to evaluate the integrals and express the solution explicitly or implicitly.
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06:18Integration by Parts for Definite Integrals
Initial Conditions and General Solutions
The general solution of a differential equation includes an arbitrary constant representing a family of solutions. Understanding how to interpret and apply initial conditions helps specify a unique solution from this family.
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05:03Initial Value Problems
Related Practice
Textbook Question
2–10. General solutions Use the method of your choice to find the general solution of the following differential equations.
y′(t) + 2y = 6
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
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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