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March 23rd, 2007, 06:46 PM   #1
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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.
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March 24th, 2007, 01:00 PM   #2
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Re: Differential Equation

z = x
so dz/dx = 2x and dy/dx = (dy/dz)(dz/dx) = 2x dy/dz,
so dy/dx = 2dy/dz + 4x dy/dz.

When x = z = 1, dy/dx = -2 so dy/dz = -1.

Substituting into the differential equation gives

2dy/dz + 4x dy/dz + (8x - 2)dy/dz + 4yx = 0,
i.e., dy/dz + 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 = (x + 1)e^(1-x).
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