Suppose that one solution y, (x) of the homogeneous second-order linear differential equation y" +p(x)y´ + q(x)y = 0 is known (on an interval I where p and q are continuous functions). The method of reduction of order consists of substituting 2(x) = v(x)y1 (x) and attempting to determine the function v(x) so that y2(x) is a second linearly independent solution of above differential equation. After substituting y = v(x)y, (x) in above differential equation, use the fact that y, (x) is a solution to deduce that y, v" + (2y,' + py,) v' = 0. If y, (x) is known, then y,v" + (2y,'+ py,) v' = 0 is a separable equation that is readily solved for the derivative v'(x) of v(x). Integration of v'(x) then gives the desired (nonconstant) function v(x). ) oft The method of reduction of order consists of substituting y, = vy, into the differential equation. Find the derivative of y2 with respect to x.
Suppose that one solution y, (x) of the homogeneous second-order linear differential equation y" +p(x)y´ + q(x)y = 0 is known (on an interval I where p and q are continuous functions). The method of reduction of order consists of substituting 2(x) = v(x)y1 (x) and attempting to determine the function v(x) so that y2(x) is a second linearly independent solution of above differential equation. After substituting y = v(x)y, (x) in above differential equation, use the fact that y, (x) is a solution to deduce that y, v" + (2y,' + py,) v' = 0. If y, (x) is known, then y,v" + (2y,'+ py,) v' = 0 is a separable equation that is readily solved for the derivative v'(x) of v(x). Integration of v'(x) then gives the desired (nonconstant) function v(x). ) oft The method of reduction of order consists of substituting y, = vy, into the differential equation. Find the derivative of y2 with respect to x.
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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