Identify the following differential equation and hence solve it y'=−4/x^2−y/x+y^2 ?

pramrok62 2022-09-29 Answered
Identify the following differential equation and hence solve it y = - 4 x 2 - y x + y 2 ?
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Answers (1)

Nolan Tyler
Answered 2022-09-30 Author has 9 answers
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 = x y y = v x
And differentiating using the product rule we get:
d v d x = x   d y d x + y d y d x = d v d x - y x
And substituting into the DE [A] we get:
d v d x - y x = - 4 x 2 - v x x + ( v x ) 2
d v d x - v x = - 4 x - v x + v 2 x
d v d x = v 2 - 4 x
Which is now separable, so we can collect terms, and "separate the variables" to get:
  1 v 2 - 4   d v =   1 x   d x
And we can integrate to get:
1 2 tanh - 1 ( v 2 ) = ln x + C
tanh - 1 ( v 2 ) = 2 ln x + 2 C
v 2 = tanh ( 2 ln x + A )
v = 2 tanh ( 2 ln x + A )
And restoring the substitution:
x y = 2 tanh ( 2 ln x + A )
y = 2 x tanh ( 2 ln x + A )
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