The differential equation d²y dx² y₁ = x² has x4 as a solution. Applying reduction order we set y2 ux 4. = Then (using the prime notation for the derivatives) y₂ = = dy 7x + 16y = 0 dx So, plugging y2 into the left side of the differential equation, and reducing, we get x²y₂ - 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
The differential equation d²y dx² y₁ = x² has x4 as a solution. Applying reduction order we set y2 ux 4. = Then (using the prime notation for the derivatives) y₂ = = dy 7x + 16y = 0 dx So, plugging y2 into the left side of the differential equation, and reducing, we get x²y₂ - 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
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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Transcribed Image Text:The differential equation
d²y
x²
dx²
dy
7x + 16y = 0
dx
has x4 as a solution.
Applying reduction order we set y2 ux 4.
=
Then (using the prime notation for the derivatives)
y₂ =
y₂ =
So, plugging y2 into the left side of the differential
equation, and reducing, we get
x²y½ − 7xy2 + 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.
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 =
solution is
and the general
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