Explanation Given: The given boundary value problem is ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = − 2 , 0 < x < π , 0 < y < π and the given boundary conditions are u ( 0 , y ) = 0 , u ( π , y ) = 0 , u ( x , 0 ) = 0 and u ( x , π ) = 0 . Calculation: The given boundary value problem is, ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 = − 2 . (1) Consider the equation, u ( x , y ) = v ( x , y ) + ψ ( x ) . (2) Partially differentiate both sides of the equation (2) with respect to x as follows. ∂ u ∂ x = ∂ v ∂ x + ψ ′ ( x ) Partially differentiate both sides of the above equation with respect to x as follows. ∂ 2 u ∂ x 2 = ∂ 2 v ∂ x 2 + ψ ″ ( x ) Substitute ∂ 2 u ∂ x 2 = ∂ 2 v ∂ x 2 + ψ ″ ( x ) and ∂ 2 u ∂ y 2 = ∂ 2 v ∂ y 2 in equation (1), ∂ 2 v ∂ x 2 + ψ ″ ( x ) + 2 = − ∂ 2 v ∂ y 2 . (3) Second derivative of ψ is zero in the equation (3) by homogeneous system, ψ ″ ( x ) = − 2 Integrate the above equation with respect to x as follows. ψ ′ ( x ) = − 2 x + c 1 Integrate the above equation with respect to x as follows. ψ ( x ) = − x 2 + c 1 x + c 2 . (4) At boundary conditions u ( 0 , y ) = 0 , u ( π , y ) = 0 , u ( x , 0 ) = 0 and u ( x , π ) = 0 , ψ ( 0 ) = 0 and ψ ( π ) = 0 . Substitute ψ ( 0 ) = 0 and ψ ( π ) = 0 in equation (4), c 1 = π and c 2 = 0 Substitute c 1 = π and c 2 = 0 in equation (4), ψ ( x ) = − x 2 + π x . (5) Consider the equation, v ( x , y ) = X ( x ) Y ( y ) . (6) Separate the variables of equation (2), X ″ X = − α 2 and Y ″ Y = − α 2 The equation of the variable X is, X ″ + α 2 X = 0 . (7) The equation of the variable Y is, Y ″ + α 2 Y = 0 . (8) The general solution of the equation (7) is, X ( x ) = c 1 cos α x + c 2 sin α x . (9) The general solution of the equation (8) is, Y ( y ) = c 3 cosh α y + c 4 sinh α y . (10) At boundary conditions u ( 0 , y ) = 0 and u ( π , y ) = 0 , X ( 0 ) = 0 X ( x ) = c 1 cos 0 + c 2 sin 0 c 1 = 0 At boundary conditions u ( 0 , y ) = 0 and u ( π , y ) = 0 , α = n Substitute c 1 = 0 and α = n in equation (9), X ( x ) = c 2 sin n x

Differential Equations with Boundary-Value Problems (MindTap Course List)

9th Edition

ISBN: 9781305965799

Author: Dennis G. Zill

Publisher: Cengage Learning

See similar textbooks

Videos

Question

Chapter 12, Problem 17RE

To determine

The solution of the boundary value problem under given boundary conditions.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12, Problem 16RE

Chapter 12, Problem 18RE

Students have asked these similar questions

Robbie Bearing Word Problems Angles name: Jocelyn date: 1/18 8K 2. A Delta airplane and an SouthWest airplane take off from an airport at the same time. The bearing from the airport to the Delta plane is 23° and the bearing to the SouthWest plane is 152°. Two hours later the Delta plane is 1,103 miles from the airport and the SouthWest plane is 1,156 miles from the airport. What is the distance between the two planes? What is the bearing from the Delta plane to the SouthWest plane? What is the bearing to the Delta plane from the SouthWest plane? Delta y SW Angles ThreeFourthsMe MATH 2

Find the derivative of the function. m(t) = -4t (6t7 - 1)6

Find the derivative of the function. y= (8x²-6x²+3)4

Chapter 12 Solutions

Differential Equations with Boundary-Value Problems (MindTap Course List)

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 solution of the boundary value problem under given boundary conditions.

Solution Summary: The author explains the solution of the boundary value problem under given boundary conditions.

Videos

The solution of the boundary value problem under given boundary conditions.

Want to see the full answer?

Chapter 12 Solutions