By M. Mimura, T. Nishida

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**The Asymptotic Solution of Linear Differential Systems: Applications of the Levinson Theorem**

The fashionable idea of linear differential structures dates from the Levinson Theorem of 1948. it's only in additional contemporary years, although, following the paintings of Harris and Lutz in 1974-7, that the importance and diversity of functions of the concept became preferred. This publication offers the 1st coherent account of the large advancements of the final 15 years.

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What we really need is a method for testing a differential equation for exactness and for constructing the corresponding function F (x, y). Both are contained in the following theorem and its proof. 5. 27) where M and N have continuous first partial derivatives at all points (x, y) in a rectangular domain D. 28) for all (x, y) in D. Remark: The proof of this theorem is rather important, as it not only provides a test for exactness, but also a method of solution for exact differential equations.

The decay law states that the amount of radioactive substance that decays is proportional at each instant to the amount of substance present. 18. The strength of a radioactive substance decreases 50% in a 30-day period. How long will it take for the radioactivity to decrease to 1% of its initial value? 19. 44 mg in 1 year. What length of time elapses before the radioactivity decreases to half its original value? 20. A tin organ pipe decays with age as a result of a chemical reaction that is catalyzed by the decayed tin.

Dx Solution We can easily verify that y1 = e2x and y2 = 5x − 5 are the solutions of the respective differential equations. Let Q1 (x) = 3e2x , Q2 (x) = 5x, and Q(x) = e2x +4x denote the right-hand sides of the three differential equations. We observe that 1 4 Q(x) = Q1 (x) + Q2 (x). 3 5 By superposition, it follows that y= 1 4 1 4 y1 + y2 = (e2x ) + (5x − 5) 3 5 3 5 is a solution of y + y = Q(x). Integrating Factor Technique In studying separable equations, we put all the terms of one variable on the left side of the equation and the terms of the other variable on the right side of the equation.