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Question

Using substitution method, find the general solution of the following homogeneous differentiation equation.

dx/dy= (y/x)/ (1+(y/x)^2)

Hence find the particular solution if y(4) =2

Many thanks!

Answer 1

dx/dy = (y/x) / (1 + (y/x)^2) ---(1)
Substitute (1) with u = y/x

dx/dy = u / (1+u^2) ---(2)

u = y/x
y = ux
d/dy (y) = d/dy (ux)
1 = x du/dy + u dx/dy
subsititute dx/dy with (2)
1 = (y/u) du/dy + u (u / (1+u^2))
1 = (y/u) du/dy + u^2 / (1+u^2)
(y/u) du/dy + u^2 / (1+u^2) - 1 = 0
(y/u) du/dy - 1 / (1+u^2) = 0
y du/dy = u / (1+u^2)
((1+u^2) / u) du = 1/y dy
(1/u + u) du = 1/y dy
ln u + 1/2 u^2 = ln y + C where C is constant

Substitute u with y/x
ln (y/x) + 1/2 (y/x)^2 = ln y + C
ln y - ln x + 1/2(y/x)^2 = ln y + C
ln x = 1/2(y/x)^2 - C
x = Ae^(1/2(y/x)^2) where A is constant

y(4) = 2,
thus
4 = Ae^(1/2(2/4)^2)
4 = Ae^(1/8)
A = 3.53

x = 3.53e^(1/2(y/x)^2)
or
x^2 = 12.46e^((y/x)^2)

To get y in terms of x
2 ln x = ln (12.46) + (y/x)^2
(y/x)^2 = 2 ln x - ln (12.46)
y^2 = x^2 [2 ln x - ln (12.46)]

Answer 2

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