Explanation The given initial value problem is y ′ ( t ) = t 2 1 − t . Take the indefinite integral of both sides of the differential equation with respect to t. ∫ y ′ ( t ) d t = ∫ ( t 2 1 − t ) d t y ( t ) = ∫ ( t 2 1 − t ) d t Obtain the indefinite integral of ∫ ( t 2 1 − t ) d t by substitution method. Let the variable v be 1 − t . Differentiate v with respect to t , d d t ( v ) = d d t ( 1 − t ) d v d t = − 1 d v = − d t Substitute v = 1 − t and d v = − d t in the integral ∫ ( t 2 1 − t ) d t as, ∫ ( t 2 1 − t ) d t = ∫ ( 1 − v ) 2 v ( − d v ) = − ∫ ( 1 − 2 v + v 2 v ) d v = − ∫ ( 1 v − 2 + v ) d v = − ( ln | v | − 2 v + v 2 2 ) + C Now replace v with 1 − t on the above expression as follows, ∫ ( t 2 1 − t ) d t = − ( ln | 1 − t | − 2 ( 1 − t ) + ( 1 − t ) 2 2 ) + C = − ln | 1 − t | + 2 ( 1 − t ) − ( 1 − t ) 2 2 + C = 2 − 2 t − ln | 1 − t | − ( 1 − t ) 2 2 + C That is, y ( t ) = 2 − 2 t − ln | 1 − t | − ( 1 − t ) 2 2 + C

Calculus: Single Variable, Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

3rd Edition

ISBN: 9780134996714

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

See similar textbooks

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…

Videos

Question

Chapter 9.1, Problem 6E

To determine

To explain: The reason for the graph of the solution of an initial value problem y′(t)=t21−t with the initial condition of y(−1)=ln2 not cross the line t=1.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.1, Problem 5E

Chapter 9.1, Problem 7E

Students have asked these similar questions

mate hat is the largest area that can be en 18 For the function y=x³-3x² - 1, use derivatives to: (a) determine the intervals of increase and decrease. (b) determine the local (relative) maxima and minima. (c) determine the intervals of concavity. (d) determine the points of inflection. b) (e) sketch the graph with the above information indicated on the graph.

use L'Hopital Rule to evaluate the following. a) 4x3 +10x2 23009×³-9 943-9 b) hm 3-84 хто бу+2 < xan x-30650)

Evaluate the next integral

Chapter 9 Solutions

Calculus: Single Variable, Early Transcendentals and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition) (Briggs, Cochran, Gillett & Schulz, Calculus Series)

Ch. 9.1 - What are the orders of the equations in Example 2?...Ch. 9.1 - Prob. 2QC Ch. 9.1 - Prob. 3QC Ch. 9.1 - Prob. 4QC Ch. 9.1 - In Example 7, if the height function were given by...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - The solution to the initial value problem y(t) = 2...

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The reason for the graph of the solution of an initial value problem y ′ ( t ) = t 2 1 − t with the initial condition of y ( − 1 ) = ln 2 not cross the line t = 1 .

Solution Summary: The author explains the graph of the solution of an initial value problem with the initial condition of y(t)=mathrmln2.

Concept explainers

Videos

To explain: The reason for the graph of the solution of an initial value problem y′(t)=t21−t with the initial condition of y(−1)=ln2 not cross the line t=1.

Want to see the full answer?

Chapter 9 Solutions