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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 1 dy dx x dz (ii) Show that equation (*) becomes = d'y dz² Suppose m₁ and m₂ represent the roots of and (iv) if m₁ and m₂ are equal, then (v) if d'y 1 d'y dx² x² dz² show that (iii) if m₁ and m₂ are real and unequal, then then = +(a − 1) du + ³y = 0 dz m² + (a1)m +B=0 m1 y(x) = c₁e₁² + c₂e²₂² = ₁x₁ + ₂x2 1 dy x² dz y(x) = (C₁+C₂2)em₁² = (c₁ + c₁ ln x)x™₁ m1 and m₂ are complex conjugate m₁ = A + iu then y(x) = x¹(c₁ cos(μ ln x) + c₂ sin(μ ln x))

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 1 dy dx x dz (ii) Show that equation (*) becomes = d'y dz² Suppose m₁ and m₂ represent the roots of and (iv) if m₁ and m₂ are equal, then (v) if d'y 1 d'y dx² x² dz² show that (iii) if m₁ and m₂ are real and unequal, then then = +(a − 1) du + ³y = 0 dz m² + (a1)m +B=0 m1 y(x) = c₁e₁² + c₂e²₂² = ₁x₁ + ₂x2 1 dy x² dz y(x) = (C₁+C₂2)em₁² = (c₁ + c₁ ln x)x™₁ m1 and m₂ are complex conjugate m₁ = A + iu then y(x) = x¹(c₁ cos(μ ln x) + c₂ sin(μ ln x))

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) + y(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 1 dy dx x dz (ii) Show that equation (*) becomes = and (iv) if m₁ and m₂ are equal, then (v) if d'y 1 d'y dx² x² dz² d'y + (x − 1) / + ³y = 0 - dz² dz Suppose m₁ and m₂ represent the roots of show that (iii) if m₁ and m₂ are real and unequal, then then = m² + (a1)m +B=0 m1 y(x) = c₁e₁² + c₂e²₂² = ₁x₁ + ₂x2 1 dy x² dz y(x) = (C₁+C₂2)em₁² = (c₁ + c₁ ln x)x™₁ m1 and m₂ are complex conjugate m₁ = A + iu then y(x) = x¹(c₁ cos(µ ln x) + c₂ sin(µ ln x)
Transcribed Image Text:1. The second order Euler equation x²y" (x)+ axy'(x) + y(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 1 dy dx x dz (ii) Show that equation (*) becomes = and (iv) if m₁ and m₂ are equal, then (v) if d'y 1 d'y dx² x² dz² d'y + (x − 1) / + ³y = 0 - dz² dz Suppose m₁ and m₂ represent the roots of show that (iii) if m₁ and m₂ are real and unequal, then then = m² + (a1)m +B=0 m1 y(x) = c₁e₁² + c₂e²₂² = ₁x₁ + ₂x2 1 dy x² dz y(x) = (C₁+C₂2)em₁² = (c₁ + c₁ ln x)x™₁ m1 and m₂ are complex conjugate m₁ = A + iu then y(x) = x¹(c₁ cos(µ ln x) + c₂ sin(µ ln x)
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