To determine The integrating factor and solve the equation [ 4 x 3 y 2 + 3 y ] + ( 3 ( x y 2 ) + 4 y ) y ′ = 0 . Explanation Theorem used: “Let the functions M , N , M y , and N x , be continuous in the rectangular region R : α < x < β , γ < y < δ . Then M ( x , y ) + N ( x , y ) y ′ = 0 is an exact differential equation in R if and only if M y ( x , y ) = N x ( x , y ) at each point of R ”. Calculation: The given equation [ 4 x 3 y 2 + 3 y ] + ( 3 ( x y 2 ) + 4 y ) y ′ = 0 can be written as follows. [ 4 x 3 y 2 + 3 y ] + ( 3 ( x y 2 ) + 4 y ) d y d x = 0 ( 4 x 3 y 2 + 3 y ) d x + ( 3 ( x y 2 ) + 4 y ) d y = 0 Here, it is observed that M = 4 x 3 y 2 + 3 y and N = 3 ( x y 2 ) + 4 y . Differentiate M = 4 x 3 y 2 + 3 y with respect to y . M y = d d y ( 4 x 3 y 2 + 3 y ) = − 4 ( − 2 ) x 3 y 3 + 3 ( − 1 ) y 2 = − 8 x 3 y 3 − 3 y 2 Differentiate N = 3 ( x y 2 ) + 4 y with respect to y . N x = d d x ( 3 ( x y 2 ) + 4 y ) = 3 y 2 By the result stated above, the equation is not exact. Compute the value of Q ( y ) = N x − M y M . N x − M y M = 3 y 2 − ( − 8 x 3 y 3 − 3 y 2 ) 4 x 3 y 2 + 3 y = 3 y 2 + 8 x 3 y 3 + 3 y 2 4 x 3 y 2 + 3 y = 6 y 2 + 8 x 3 y 3 4 x 3 y 2 + 3 y = 6 y + 8 x 3 y 3 4 x 3 + 3 y y 2 Compute the integrating factor

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 2.6, Problem 30P

To determine

The integrating factor and solve the equation [4x3y2+3y]+(3(xy2)+4y)y′=0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 2.6, Problem 29P

Chapter 2.6, Problem 31P

Students have asked these similar questions

Let G be a graph with n ≥ 2 vertices x1, x2, . . . , xn, and let A be the adjacency matrixof G. Prove that if G is connected, then every entry in the matrix A^n−1 + A^nis positive.

Module Code: MATH380202 1. (a) Define the terms "strongly stationary" and "weakly stationary". Let {X} be a stochastic process defined for all t € Z. Assuming that {X+} is weakly stationary, define the autocorrelation function (acf) Pk, for lag k. What conditions must a process {X+) satisfy for it to be white noise? (b) Let N(0, 1) for t€ Z, with the {+} being mutually independent. Which of the following processes {X+} are weakly stationary for t> 0? Briefly justify your answers. i. Xt for all > 0. ii. Xo~N(0,) and X₁ = 2X+-1+ &t for t > 0. (c) Provide an expression for estimating the autocovariance function for a sample X1,..., X believed to be from a weakly stationary process. How is the autocor- relation function Pk then estimated, and a correlogram (or acf plot) constructed? (d) Consider the weakly stationary stochastic process ✗+ = + + +-1+ +-2 where {E} is a white noise process with variance 1. Compute the population autocorre- lation function Pk for all k = 0, 1, ....

iii) i=5 x² = Σ i=1 (Yi — mi)² σ 2 By minimising oc², derive the formulae for the best values of the model for a 1 degree polynomial (2 parameters).

Chapter 2 Solutions

Elementary Differential Equations

Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 3P Ch. 2.1 - Prob. 4P Ch. 2.1 - Prob. 5P Ch. 2.1 - Prob. 6P Ch. 2.1 - In each of Problems 1 through 12: Draw a direction...Ch. 2.1 - Prob. 8P Ch. 2.1 - Prob. 9P Ch. 2.1 - Prob. 10P

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.