Explanation Procedure used: “Solution of system of differential equations by elimination method. 1 Consider the system of differential equations. 2 Express the equations in differential operator form. 3 Find the DE by elimination. 4 Obtain the complementary function. 5 Further, Obtain the particular solution. 6 Ultimately write the general solution in form y ( x ) = y c + y p .” Calculation: Consider the system of differential equations d x d t + d y d t = e t ...... ( 1 ) − d 2 x d t 2 + d x d t + x + y = 0 ..... ( 2 ) . Write the equations in differential operator form. D x + D y = e t … ( 1 ) ( − D 2 + D + 1 ) x + y = 0 … ( 2 ) × D To eliminate y . Multiply ( 2 ) × D . Subtract ( 1 ) − ( 2 ) . D x + D y − D ( − D 2 + D + 1 ) x − D y = e t D x − ( − D 3 + D 2 + D ) x = e t ( D 3 − D 2 ) x = e t Find the roots of auxiliary equation of the homogeneous equation associate to nonhomogeneous DE ( D 3 − D 2 ) x = e t . m 3 − m 2 = 0 m 2 ( m − 1 ) = 0 m 1 = 0 , m 2 = 0 and m 3 = 1 Thus, the complementary function is x c = c 1 + c 2 t + c 3 e t . The right-hand side of the DE is a polynomial. Therefore, consider the particular solution in form x p = A t e t . Substitute x p = A t e t , x ′ p = A e t + A t e t , x ″ p = 2 A e t + A t e t and x ‴ p = 3 A e t + A t e t into the DE. 3 A e t + A t e t − 2 A e t − A t e t = e t A e t = e t A = 1 Hence, the particulars solution is x p = t e t

Differential Equations with Boundary-Value Problems

9th Edition

ISBN: 9781337632515

Author: Dennis G. Zill

Publisher: Cengage Learning US

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Chapter 4.9, Problem 14E

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To solve: The system of differential equations dxdt+dydt=et,−d2xdt2+dxdt+x+y=0 by systematic elimination.

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Chapter 4 Solutions

Differential Equations with Boundary-Value Problems

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The system of differential equations d x d t + d y d t = e t , − d 2 x d t 2 + d x d t + x + y = 0 by systematic elimination.

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To solve: The system of differential equations dxdt+dydt=et,−d2xdt2+dxdt+x+y=0 by systematic elimination.

Chapter 4 Solutions