Identify the following differential equation and hence solve it #y'=-4/x^(2)-y/x+y^(2)# ?
1 Answer
# y =2/xtanh(2lnx + A)#
Explanation:
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)#
