What is the general solution of the differential equation dy/dx=(x+y)/x?

trkalo84

trkalo84

Answered question

2022-09-18

What is the general solution of the differential equation d y d x = x + y x ?

Answer & Explanation

Kelbelol

Kelbelol

Beginner2022-09-19Added 10 answers

If we use the suggested substitution, y=vx then differentiating wrt x and applying the product rule we have:

d y d x = ( v ) ( d d x x ) + ( d d x v ) ( x )
          = ( v ) ( 1 ) + d v d x x
          = v + x d v d x

Substituting this result into the initial differential equation [A] we get:

v + x d v d x = x + v x x
v + x d v d x = 1 + v
x d v d x = 1
d v d x = 1 x

Which has reduced the equation to a trivial First Order separable equation, which we can "separate the variables" to get:

  d v =   1 x   d x

And if we integrate we get:

v = ln | x | + C

Restoring the earlier substitution, we get:

y x = ln | x | + C

Leading to the General Solution:

y = x ln | x | + C x

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