Answer to Question #150862 in Differential Equations for Ali

Question

A second order homogenous differential equation with constant coefficient and polynomial
coefficient is given, apply Euler-Cauchy equation to solve the given differential equation by
using the provided initial value problem (IVP).
x^2 y'' +8y' +10y =0 y(1) = 0 , y'(1) = 2.5

Answer 1

"\\displaystyle\nx^2 y'' +8xy' +10y =0,\\,\\, y(1) = 0,\\,\\, y'(1) = 2.5\\\\\n\n\\textsf{Let}\\,\\,y = f(\\ln{x}), y' = \\frac{f'(\\ln{x})}{x}\\\\\n\ny'' = \\frac{f''(\\ln{x})}{x^2} - \\frac{f'(\\ln{x})}{x^2}\\\\\n\nf"(\\ln{x}) - f'(\\ln{x}) + \\frac{8f'(\\ln{x})}{x}\\times x + 10f(\\ln{x}) = 0\\\\\n\nf"(\\ln{x}) - f'(\\ln{x}) + 8f'(\\ln{x}) + 10f(\\ln{x}) = 0\\\\\n\nf"(\\ln{x}) + 7f'(\\ln{x}) + 10f(\\ln{x}) = 0\\\\\n\n\\textsf{Let}\\,\\, t = \\ln{x}\\\\\n\nf"(t) + 7f'(t) + 10f(t) = 0\\\\\n\n\n\\textsf{The auxiliary equation is}\\\\\n\nm^2 + 7m + 10 = 0\\\\\n\n(m + 5)(m + 2) = 0\\\\\n\n\nm = -2, -5\\\\\n\n\nf(t) = Ae^{-2t} + Be^{-5t}\\,\\, \\textsf{is the general solution}\\\\\n\nf(\\ln{x}) = Ae^{-2\\ln{x}} + Be^{-5\\ln{x}}\\\\\n\n\ny(x) = f(\\ln{x}) = \\frac{A}{x^2} + \\frac{B}{x^5}\\\\\n\n\ny(1) = A + B = 0\\\\\n\n\ny'(x) = -2\\frac{A}{x^3} - 5\\frac{B}{x^6}\\\\\n\ny'(1) = -2A - 5B = \\frac{5}{2}\\\\\n\n\n\\textsf{But}\\,\\, A = -B\\\\\n\n-2A + 5A = \\frac{5}{2}\\\\\n\n\n3A = \\frac{5}{2}\\\\\n\nA = \\frac{5}{6}, B = -\\frac{5}{6}\\\\\n\n\ny(x) = \\frac{5}{6x^2} - \\frac{5}{6x^5}"