Identify the following differential equation and hence solve it #y'=-4/x^(2)-y/x+y^(2)# ?

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Answer 1 · 2018-07-01T21:48:07

# y =2/xtanh(2lnx + A)#

We have:

# y' = -4/x^2-y/x+y^2 # ..... [A]

This is a non-linear first order Differential Equation. We can attempt a substitution:

# v = xy iff y=v/x #

And differentiating using the product rule we get:

# (dv)/(dx) = x \ dy/dx + y => dy/dx = ((dv)/(dx)-y)/x #

And substituting into the DE [A] we get:

# ((dv)/(dx)-y)/x = -4/x^2-(v/x)/x+(v/x)^2 #

# :. (dv)/(dx)-v/x = -4/x-v/x+v^2/x #

# :. (dv)/(dx) = (v^2-4)/x #

Which is now separable, so we can collect terms, and "separate the variables" to get:

# int \ 1/(v^2-4) \ dv = int \ 1/x \ dx #

And we can integrate to get:

# 1/2tanh^(-1)(v/2) =lnx + C#

# :. tanh^(-1)(v/2) =2lnx + 2C#

# :. v/2 =tanh(2lnx + A)#

# :. v =2tanh(2lnx + A)#

And restoring the substitution:

# xy =2tanh(2lnx + A)#

# y =2/xtanh(2lnx + A)#