Explanation Given: System of differential equations: d 2 x d t 2 − x + y = 0 , ( 1 ) x + d 2 y d t 2 − y = e 3 t ( 2 ) Approach: 1) Use elimination method for solving algebraic systems. Step1: Add or subtract the given equations to get an equation in single variable. Step2: Solve the single variable equation. Step3: Substitute the value of variable obtained in Step2 in other equations to get the values of other unknowns. 2) Use method of undetermined coefficients to find particular solution of non homogeneous ordinary differential equation. In this method, find the form of particular solution and then solve according to that. Calculation: Write Equation ( 2 ) in operator form. ( D 2 − 1 ) x + y = 0 ( 3 ) x + ( D 2 − 1 ) y = e 3 t ( 4 ) Multiply Equation ( 3 ) with ( 1 − D 2 ) . ( 1 − D 2 ) x − ( D 2 − 1 ) 2 y = ( 1 − D 2 ) e 3 t ( 1 − D 2 ) x − ( D 2 − 1 ) 2 y = − 8 e 3 t ( 5 ) Add Equation ( 3 ) and Equation ( 5 ) . ( 1 − ( D 2 − 1 ) 2 ) y = − 8 e 3 t ( 6 ) The auxiliary equation of Equation ( 6 ) is, 1 − ( r 2 − 1 ) 2 = 0 r 2 ( r 2 − 2 ) = 0 ( 7 ) Roots of Equation ( 7 ) are r = 0 , 0 , ± 2 . So, the complimentary solution of Equation ( 6 ) is y h ( t ) = A + B t + C e 2 t + D e − 2 t Here, A , B , C and D are arbitrary constants. The particular solution of Equation ( 6 ) takes the form y p = k e 3 t . ( 8 ) Differentiate Equation ( 8 ) with respect to t. y ′ p = 3 k e 3 t ` ( 9 ) Differentiate Equation ( 9 ) with respect to t. y ″ p = 9 k e 3 t ( 10 ) Differentiate Equation ( 10 ) with respect to t

Fundamentals of Differential Equations and Boundary Value Problems

7th Edition

ISBN: 9780321977106

Author: Nagle, R. Kent

Publisher: Pearson Education, Limited

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Chapter 6.3, Problem 39E

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The general solution x(t) and y(t).for the systems of differential equations.

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Chapter 6.3, Problem 38E

Chapter 6.4, Problem 1E

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...

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The general solution x ( t ) and y ( t ) .for the systems of differential equations.

Solution Summary: The author explains the elimination method for solving algebraic systems, and the method of undetermined coefficients to find particular solution of non homogeneous ordinary differential equations.

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To find:

The general solution x(t) and y(t).for the systems of differential equations.

Want to see the full answer?

Chapter 6 Solutions

The general solution x ( t ) and y ( t ) .for the systems of differential equations.

Solution Summary: The author explains the elimination method for solving algebraic systems, and the method of undetermined coefficients to find particular solution of non homogeneous ordinary differential equations.

Videos

To find: The general solution x(t) and y(t).for the systems of differential equations.

Want to see the full answer?

Chapter 6 Solutions

To find:

The general solution x(t) and y(t).for the systems of differential equations.