Explanation Given: The Cauchy-Euler equation, x 3 y ‴ − 3 x y ′ + 3 y = x 4 cos x ... ( 1 ) x > 0 . Approach: To determine the general solution of the equation first find the complimentary function of the given equation. Then find out four Wronskian of determinants. Use the formula to find the particular solution of the equation. y p ( x ) = ∑ k = 1 n y k ( x ) ∫ g ( x ) W K ( x ) W [ y 1 , ... , y n ] ( x ) d x The general solution of the equation is, y = y c + y p . The general equation of complementary function is, C F = c 1 y 1 ( x ) + c 2 y 2 ( x ) + c 3 y 3 ( x ) ... ( 2 ) Calculation: Consider, y = x r y ′ = r ( x r − 1 ) y ″ = r ( r − 1 ) x r − 2 y ‴ = r ( r − 1 ) ( r − 2 ) x r − 3 Substitute the above value in equation ( 1 ) . x 3 ( r ( r − 1 ) ( r − 2 ) ) x r − 3 − 3 x ( r x r − 1 ) + 3 x r = 0 ( r ( r − 1 ) ( r − 2 ) ) x r − 3 r x r + 3 x r = 0 [ ( r ( r − 1 ) ( r − 2 ) ) − 3 r + 3 ] x r = 0 ( r ( r − 1 ) ( r − 2 ) ) − 3 r + 3 = 0 ( r 2 − r ) ( r − 2 ) − 3 r + 3 = 0 r 2 ( r − 2 ) − r ( r − 2 ) − 3 r + 3 = 0 r 3 − 2 r 2 − r 2 + 2 r − 3 r + 3 = 0 r 3 − 3 r 2 − r + 3 = 0 ( r − 1 ) ( r 2 − 2 r − 3 ) = 0 ( r − 1 ) ( r − 3 ) ( r + 1 ) = 0 r = 1 , − 1 , 3 So, the fundamental set of the homogeneous equation is, { x r ( r = 1 , − 1 , 3 ) } = { x , x − 1 , x 3 } . The complimentary function is, C F = c 1 x + c 2 x − 1 + c 3 x 3 . Compare the above Equation with Equation ( 2 ) . y 1 ( x ) = x y 2 ( x ) = x − 1 And, y 3 ( x ) = x 3 . Write four Wronskian of determinant. W [ y 1 , y 2 , y 3 ] ( x ) = | y 1 y 2 y 3 y ′ 1 y ′ 2 y ′ 3 y ″ 1 y ″ 2 y ″ 3 | = | x x − 1 x 3 1 − x − 2 3 x 2 0 2 x − 3 6 x | = x ( − 6 x − 1 − 6 x − 1 ) − x − 1 ( 6 x − 0 ) + x 3 ( 2 x − 3 − 0 ) = − 12 − 6 + 2 And, W [ y 1 , y 2 , y 3 ] ( x ) = − 16 Multiply the equation ( 1 ) by 1 x 3 . ( x 3 y ‴ − 3 x y ′ + 3 y ) 1 x 3 = ( x 4 cos x ) 1 x 3 y ‴ − 3 x 2 y ′ + 3 x 3 y = x cos x Consider, g ( x ) = x cos x . Now, the value of W 1 , W 2 and W 3 are, W 1 ( x ) = ( − 1 ) 3 − 1 W [ y 2 , y 3 , ] ( x ) = ( − 1 ) 2 | x − 1 x 3 − x − 2 3 x 2 | = ( 1 ) [ ( x − 1 ) ( 3 x 2 ) − ( − x − 2 ) ( x 3 ) ] = 3 x + x And, W 1 ( x ) = 4 x Similarly, W 2 follow same procedure. W 2 ( x ) = ( − 1 ) 3 − 2 | x x 3 1 3 x 2 | W 2 ( x ) = − ( 3 x 3 − x 3 ) W 2 ( x ) = − 2 x 3 Similarly W 3 is written as, W 3 ( x ) = ( − 1 ) 3 − 3 | x x − 1 1 − x − 2 | = x − 1 − x − 1 = 2 x − 1 Now particular solution is written as with the help of formula. y p ( x ) = ∑ k = 1 n y k ( x ) ⋅ v k ( x ) , k = 1 , ... , n y p ( x ) = y 1 ( x ) ⋅ v 1 ( x ) + y 2 ( x ) ⋅ v 2 ( x ) + y 3 ( x ) ⋅ v 3 ( x )

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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 6.4, Problem 11E

To determine

The general solution to the given Cauchy-Euler equation x3y‴−3xy′+3y=x4cosx, x>0.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 6.4, Problem 10E

Chapter 6.4, Problem 12E

Students have asked these similar questions

5.15 The ends of the laminated-wood roof arch shown are tied together with a horizontal steel rod 90 feet, "10 inches" long, which must withstand a total load of 60 k. Two turn- buckles are used. All threaded rod ends are upset. a. Determine the required diameter D of the rod if the maximum allowable stress is 20 ksi. b. If the unstressed length of the rod is 90 feet, "10 inches" and there are four threads per inch on the upset ends, how many turns of one turnbuckle will bring it back to its unstressed length after it has elongated under full allowable tensile stress? E = 29 × 103 ksi. a. Required diameter D b. Number of turns LAMINATED WOOD ROOF ARCH TIE ROD TURNBUCKLE DETAIL: UPSET END

1. 2. In Exercises 1, 2, 3, 4, 5, 6, 7, and 8, use the graph of the function f to find lim f(x), lim f(x), and xa x+a+ lim f(x) at the indicated value of a, if the limit exists. x a Answ 2: the init X1st

5.8 A reinforced concrete column is 12 feet long, and un- der load, it shortens 3 inches". Determine its average unit strain. Average Unit Strain

Chapter 6 Solutions

Fundamentals of Differential Equations and Boundary Value Problems

Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - In Problems 1-6, determine the largest interval...Ch. 6.1 - Prob. 7E Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...Ch. 6.1 - In Problems7-14, determine whether the given...

Knowledge Booster

Learn more about

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

The general solution to the given Cauchy-Euler equation x 3 y ‴ − 3 x y ′ + 3 y = x 4 cos x , x > 0 .

Solution Summary: The author explains how to determine the general solution of the Cauchy-Euler equation.

Concept explainers

Videos

The general solution to the given Cauchy-Euler equation x3y‴−3xy′+3y=x4cosx, x>0.

Want to see the full answer?

Chapter 6 Solutions

The general solution to the given Cauchy-Euler equation x 3 y ‴ − 3 x y ′ + 3 y = x 4 cos x , x &gt; 0 .

Solution Summary: The author explains how to determine the general solution of the Cauchy-Euler equation.

Concept explainers

Videos

The general solution to the given Cauchy-Euler equation x3y‴−3xy′+3y=x4cosx, x>0.

Want to see the full answer?

Chapter 6 Solutions

The general solution to the given Cauchy-Euler equation x 3 y ‴ − 3 x y ′ + 3 y = x 4 cos x , x > 0 .