Explanation Given information: The given expressions are d x d t = x + y , d y d t = x 2 − y 2 . Calculation: First order ODE which can be solved use the integrating factor method. If a differential equation is in the form d x d y + p ( y ) x = Q ( y ) Integrating factor, μ ( y ) = exp ( ∫ P ( y ) d x ) The general solution is, μ ( y ) x ( y ) = ∫ ( Q ( y ) ⋅ μ ( y ) ) d y + C The objective is to function x = x ( t ) and y = y ( t ) which satisfy the system of linear equation. Consider the differential equation, d x d t = x + y …. (1) d y d t = x 2 − y 2 .... (2) From above parametric differential equation, d x d t = x + y And d y d t = ( x + y ) ( x − y ) d y d x = d y d t ⋅ d t d x d y d x = x − y d y d x + y = x From the equation d y d x + y = x Integrating factor, μ ( t ) = e ∫ 1 d x Integrate with respect to x , μ ( t ) = e x The General solution is y e x = ∫ x e x d x Use integration by parts ∫ u v d

6th Edition

ISBN: 9781465208880

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Chapter 14, Problem 44SP

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To find: The functions that satisfy the linear equation.

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Chapter 14, Problem 43SP

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