Exercise 2. (6 points) (a) Give the definition of a solution to a first order linear differential equation. (b) Find all solutions to the differential equation y'(x) ex y(x) = and identify the domain for each of х X them. (c) Does there exist a solution y(x) such that lim y(x) exists and is finite? Justify your answer. (a) Let x→0+ y' = a(xy + blx) be a first order linear differential where a, b are continuous functions on an interval ICB. A solution is a function y E C'(I) such that g'(x) = a (2)y(2) + b (x) for x ε I +b(x) YE (b) The given equation is a first order linear differential equation with coefficients a(x) = −11, b(x) = x, defined for x 0. We may thus search for solutions in the two intervals (-∞0, 0) and (0, +∞). If we pick a primitive y = A(x) of the function y = a(x) and use the formula [b(x)e¯ b(x)e-A(x) dr y = eA(x) we obtain the two families of solutions e+c y(x) , x y(x): = ex + d X for x 0, Vc Є R, for x < 0, Vd E R. (c) Writing the limit lim y(x) = ex + c = x+0+ lim x+0+ X = lim x+0+ 1+x+o(x)+c X and varying c ER, we obtain that the limit exists finitely and has value 1 if and only if c = -1. For all c-1 the limit is not finite.

Exercise 2. (6 points) (a) Give the definition of a solution to a first order linear differential equation. (b) Find all solutions to the differential equation y'(x) ex y(x) = and identify the domain for each of х X them. (c) Does there exist a solution y(x) such that lim y(x) exists and is finite? Justify your answer. (a) Let x→0+ y' = a(xy + blx) be a first order linear differential where a, b are continuous functions on an interval ICB. A solution is a function y E C'(I) such that g'(x) = a (2)y(2) + b (x) for x ε I +b(x) YE (b) The given equation is a first order linear differential equation with coefficients a(x) = −11, b(x) = x, defined for x 0. We may thus search for solutions in the two intervals (-∞0, 0) and (0, +∞). If we pick a primitive y = A(x) of the function y = a(x) and use the formula [b(x)e¯ b(x)e-A(x) dr y = eA(x) we obtain the two families of solutions e+c y(x) , x y(x): = ex + d X for x 0, Vc Є R, for x < 0, Vd E R. (c) Writing the limit lim y(x) = ex + c = x+0+ lim x+0+ X = lim x+0+ 1+x+o(x)+c X and varying c ER, we obtain that the limit exists finitely and has value 1 if and only if c = -1. For all c-1 the limit is not finite.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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