Explanation Given: The second order differential equation 2 y ″ − 3 y ′ + y = 0 and y ( 0 ) = 2 , y ′ ( 0 ) = 1 2 . Formula used: The auxiliary equation of a linear homogeneous differential equation of the form a 0 y ″ + a 1 y ′ + a 2 y = 0 with constant coefficients is given by a 0 r 2 + a 1 r + a 2 = 0 . The general solution of the auxiliary equation with complex roots is given by y = e α x ( c 1 sin β x + c 2 cos β x ) . The general solution of the auxiliary equation with double roots r is given by y = e r x ( c 1 + c 2 x ) . The general solution of the auxiliary equation with two distinct roots r 1 and r 2 , is given by y = c 1 e r 1 x + c 2 e r 2 x . Calculation: The given differential equation is 2 y ″ − 3 y ′ + y = 0 . Compare the given equation with a 0 y ″ + a 1 y ′ + a 2 y = 0 and obtain the values of a 0 , a 1 , a 2 as shown below: a 0 = 2 , a 1 = − 3 , a 2 = 1 . Write the auxiliary equation of the homogeneous equation 2 y ″ − 3 y ′ + y = 0 as, 2 r 2 − 3 r + 1 = 0 Solve the auxiliary equation and obtain the roots as follows: 2 r 2 − 3 r + 1 = 0 ( r − 1 2 ) ( r − 1 ) = 0 r = 1 2 , 1 Thus, the auxiliary equation has three distinct roots r = 1 2 , 1 . Using the above formula, obtain the general solution of the homogeneous equation 2 y ″ − 3 y ′ + y = 0 as, y = c 1 e r 1 t + c 3 e r 2 t y = c 1 e t 2 + c 2 e t Therefore, the general solution of the differential equation is y = c 1 e t 2 + c 2 e t

10th Edition

ISBN: 9780470458327

Author: William E. Boyce, Richard C. DiPrima

Publisher: Wiley, John & Sons, Incorporated

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 3.1, Problem 20P

To determine

The solution of the initial value problem and determine the maximum value and the point where the solution is zero.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.1, Problem 19P

Chapter 3.1, Problem 21P

Students have asked these similar questions

Let U = = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Use the following subsets of U to determine if each statement is true or false. A = {0, 1, 3, 5} and B = {2, 3, 4, 5,9} • true AUB = {3,5} • true A - B = {0, 1} ⚫ true B = {0, 1, 6, 7, 8, 10} ⚫ true An Bc • true (AUB) = {0,1} = {0, 1, 2, 4, 6, 7, 8, 9, 10} ⚫ true A x B = {(0,2), (1, 3), (3, 4), (5,5)}

Let A = {x Z | x=0 (mod 6)} and B = {x = Z | x = 0 (mod 9)}. Which of the following sentences describes the set relationship between A and B ? *Keep in mind that Ç means proper subset. AÇ B BÇA A = B AnB = 0 none of these

Let U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} be the universal set. Let A = {0, 1, 2, 3, 9} and B = {2, 3, 4, 5, 6}. Select all elements in An B. 2 3 4 5 18 7 8 9 ☐ 10

Chapter 3 Solutions

Elementary Differential Equations

Knowledge Booster

$Advanced Math$

Learn more about

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