The differential equation x2 dx2 d²y dy 7x- +16y=0 dx has x4 as a solution. Applying reduction order we set y2 = ux¹. Then (using the prime notation for the derivatives) y2 = y₁₂ = So, plugging y2 into the left side of the differential equation, and reducing, we get x²y 7xy2+16y2 The reduced form has a common factor of x5 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 = u' giving us 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
The differential equation x2 dx2 d²y dy 7x- +16y=0 dx has x4 as a solution. Applying reduction order we set y2 = ux¹. Then (using the prime notation for the derivatives) y2 = y₁₂ = So, plugging y2 into the left side of the differential equation, and reducing, we get x²y 7xy2+16y2 The reduced form has a common factor of x5 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 = u' giving us 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
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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Transcribed Image Text:The differential equation
x2
dx2
d²y
dy
7x-
+16y=0
dx
has x4
as a solution.
Applying reduction order we set y2 = ux¹.
Then (using the prime notation for the derivatives)
y2 =
y₁₂ =
So, plugging y2 into the left side of the differential equation, and
reducing, we get
x²y 7xy2+16y2
The reduced form has a common factor of x5 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 = u' giving us 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
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