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22 d'y dx² dy 7x- +16y=0 dx

22 d'y dx² dy 7x- +16y=0 dx

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1.5
1.5) The differential equation d²y dy dx² da 7x- +16y=0 has x4 as a solution. Applying reduction order we set y₂ = Ux¹. Then (using the prime notation for the derivatives) 3/2= y So, plugging y2 into the left side of the differential equation, and reducing, we get x²y — 7xy + 16y₂ ||||| The reduced form has a common factor of 5 which we can divide out of the equation so that we have xu" +u = 0. u' giving us Since this equation does not have any u terms in it we can make the substitution w = the first order linear equation xw' + w = 0. 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 = Finally y2 = and the general solution is Jam 262
Transcribed Image Text:1.5) The differential equation d²y dy dx² da 7x- +16y=0 has x4 as a solution. Applying reduction order we set y₂ = Ux¹. Then (using the prime notation for the derivatives) 3/2= y So, plugging y2 into the left side of the differential equation, and reducing, we get x²y — 7xy + 16y₂ ||||| The reduced form has a common factor of 5 which we can divide out of the equation so that we have xu" +u = 0. u' giving us Since this equation does not have any u terms in it we can make the substitution w = the first order linear equation xw' + w = 0. 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 = Finally y2 = and the general solution is Jam 262
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