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r²y" (r) - 2ry'(x) + 2y(x) = 3x² + 2lnr 1

r²y" (r) - 2ry'(x) + 2y(x) = 3x² + 2lnr 1

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. The second order Euler equation x²y" (x) + axy(x) + By(x) = 0 can be reduced to aa second order linear equation with consant coefficient by appropriate change of the independent variaable. (i) Show that dy dx (ii) Show that equation (*) becomes 1 dy z dz and d'y d₂² Suppose m₁ and m₂ represent the roots of (iv) if m₁ and m₂ are equal, then dy +(a-1)+By = 0 dz d'y dx² show that (iii) if m₁ and m₂ are real and unequal, then then m² + (a1)m+B=0 1 d'y ² dz² y(x) = c₁e₁+c₂e₂x = ₁2₁ +₂22 y(x) = (₁ +₂2)em₁ = (₁+c₁ lnr)r™ (v) if m₁ and m₂ are complex conjugate m₁ = A + iu then 2. Use the previous method in question (1), solve 1 dy r² dz y(x) = 2¹ (c₁ cos(μ ln x) + c₂ sin(μln x)) 1 x²y'(x) - 2xy' (x) + 2y(x) = 3x²+2 ln x
Transcribed Image Text:1. The second order Euler equation x²y" (x) + axy(x) + By(x) = 0 can be reduced to aa second order linear equation with consant coefficient by appropriate change of the independent variaable. (i) Show that dy dx (ii) Show that equation (*) becomes 1 dy z dz and d'y d₂² Suppose m₁ and m₂ represent the roots of (iv) if m₁ and m₂ are equal, then dy +(a-1)+By = 0 dz d'y dx² show that (iii) if m₁ and m₂ are real and unequal, then then m² + (a1)m+B=0 1 d'y ² dz² y(x) = c₁e₁+c₂e₂x = ₁2₁ +₂22 y(x) = (₁ +₂2)em₁ = (₁+c₁ lnr)r™ (v) if m₁ and m₂ are complex conjugate m₁ = A + iu then 2. Use the previous method in question (1), solve 1 dy r² dz y(x) = 2¹ (c₁ cos(μ ln x) + c₂ sin(μln x)) 1 x²y'(x) - 2xy' (x) + 2y(x) = 3x²+2 ln x
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