d'y dy da 7x- + 16y = 0 da2 has r as a solution. Applying reduction order we set y2 = ux". Then (using the prime notation for the derivatives) So, plugging y2 into the left side of the differential equation, and reducing, we get x'y-7xy, + 16y2 = The reduced form has a common factor of x° 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 rw + 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
d'y dy da 7x- + 16y = 0 da2 has r as a solution. Applying reduction order we set y2 = ux". Then (using the prime notation for the derivatives) So, plugging y2 into the left side of the differential equation, and reducing, we get x'y-7xy, + 16y2 = The reduced form has a common factor of x° 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 rw + 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
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:dy
dy
+ 16y = 0
da
4
has x as a solution.
Applying reduction order we set y2 = ux
Then (using the prime notation for the derivatives)
So, plugging y, into the left side of the differential equation, and reducing, we get
7xy, + 16y2
18338
The reduced form has a common factor of x° 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.
for x > 0.
This equation has integrating factor
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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