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The differential equation 4 has x as a solution. Applying reduction order we set y = ux¹. Then (using the prime notation for the derivatives) d²y dy - 7x- +16y=0 dz² dx Y = 7c1x^6 y" = So, plugging y₂ into the left side of the differential equation, and reducing, we get x²y - 7xy + 16y2 = = u' giving us the first order linear equation xw' + w = 0. The reduced form has a common factor of 25 which we can divide out of the equation so that we have xu" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w = Integrating to get u, and using b as our second constant of integration we have u = and the general solution is Finally₂ =

The differential equation 4 has x as a solution. Applying reduction order we set y = ux¹. Then (using the prime notation for the derivatives) d²y dy - 7x- +16y=0 dz² dx Y = 7c1x^6 y" = So, plugging y₂ into the left side of the differential equation, and reducing, we get x²y - 7xy + 16y2 = = u' giving us the first order linear equation xw' + w = 0. The reduced form has a common factor of 25 which we can divide out of the equation so that we have xu" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w = Integrating to get u, and using b as our second constant of integration we have u = and the general solution is Finally₂ =

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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The differential equation 4 has x as a solution. Applying reduction order we set y = ux¹. Then (using the prime notation for the derivatives) d²y dy - 7x- +16y=0 dz² dx Y = 7c1x^6 y" = So, plugging y₂ into the left side of the differential equation, and reducing, we get x²y - 7xy + 16y2 = = u' giving us the first order linear equation xw' + w = 0. The reduced form has a common factor of 25 which we can divide out of the equation so that we have xu" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w = Integrating to get u, and using b as our second constant of integration we have u = and the general solution is Finally₂ =
Transcribed Image Text:The differential equation 4 has x as a solution. Applying reduction order we set y = ux¹. Then (using the prime notation for the derivatives) d²y dy - 7x- +16y=0 dz² dx Y = 7c1x^6 y" = So, plugging y₂ into the left side of the differential equation, and reducing, we get x²y - 7xy + 16y2 = = u' giving us the first order linear equation xw' + w = 0. The reduced form has a common factor of 25 which we can divide out of the equation so that we have xu" + u' = 0. Since this equation does not have any u terms in it we can make the substitution w This equation has integrating factor for x > 0. If we use a as the constant of integration, the solution to this equation is w = Integrating to get u, and using b as our second constant of integration we have u = and the general solution is Finally₂ =
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