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March 23rd, 2007, 05:46 PM  #1 
Newbie Joined: Dec 2006 Posts: 9 Thanks: 0  Differential Equation
Using the substitution x=z^(1/2), transform the differential equation d^2y/dx^2 + (4x  1/x)dy/dx + 4yx^2 = 0 into one relating y and z. hence find y in terms of x given that dy/dx = 2 and y = 2 when x=1. 
March 24th, 2007, 12:00 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,467 Thanks: 2038  Re: Differential Equation
z = x² so dz/dx = 2x and dy/dx = (dy/dz)(dz/dx) = 2x dy/dz, so d²y/dx² = 2dy/dz + 4x² d²y/dz². When x = z = 1, dy/dx = 2 so dy/dz = 1. Substituting into the differential equation gives 2dy/dz + 4x² d²y/dz² + (8x²  2)dy/dz + 4yx² = 0, i.e., d²y/dz² + 2dy/dz + y = 0. Multiplying by e^z and integrating gives e^z dy/dz + e^zy = e (since dy/dz = 1 and y = 2 when x = z = 1). Integrating again gives e^z y = ez + e (since y = 2 when x = z = 1). Hence y = (x² + 1)e^(1x²). 

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