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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
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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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 Y2(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). The method of reduction of order consists of substituting y2 = vy, into the differential equation. Find the derivative of y2 with respect to x. Y2' =D
Transcribed Image Text: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 Y2(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). The method of reduction of order consists of substituting y2 = vy, into the differential equation. Find the derivative of y2 with respect to x. Y2' =D
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