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A step-by-step guide on how to use the Reduction of Order method to find a second solution for the given differential equation: x2y′′ −2y = 0, with y1 = x2. the calculations and discussions on the process.
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Bernd Schr¨oder
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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∫ (^) e− ∫^ P dx y^21 dx,^ which is obtained by doing the above substitution symbolically.
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2 y′^ = u′x^2 + u 2 x
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2
u′′x^2 + u′ 4 x + u 2
− 2 ux^2 = 0
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2
u′′x^2 + u′ 4 x + u 2
− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2
u′′x^2 + u′ 4 x + u 2
− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 = 0 u′′x + u′ 4 = 0
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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y = ux^2 y′^ = u′x^2 + u 2 x y′′^ = u′′x^2 + u′ 4 x + u 2 x^2
u′′x^2 + u′ 4 x + u 2
− 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 + u 2 x^2 − 2 ux^2 = 0 u′′x^4 + u′ 4 x^3 = 0 u′′x + u′ 4 = 0 v := u′
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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dv dx =^ −
4 v x
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science
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dv dx =^ −
4 v x dv v =^ −^4
dx x
Bernd Schr¨oder Louisiana Tech University, College of Engineering and Science