I was given a problem, it has been worded as follows: Use the substitution z = y

deformere692qr 2022-05-16 Answered
I was given a problem, it has been worded as follows: Use the substitution z = y x to transform the differential equation d y d x = 3 x y y 2 3 x 2 , into a linear equation. Hence obtain the general solution of the original equation.
Workings:
d y d x = 3 ( y x ) ( y x ) 2 3
Multiplied by 1 x 2 1 x 2
d y d x = 3 z z 2 3
z = y x y x 2
( z + y x 2 ) x = 3 z z 2 3
After a whole page of workings, I arrive at this
z + z x = 3 z
Technically, this is a linear equation since it takes the form y'+p(x)y=f(x) . Since z=f(y,x). However, I'm unable to compute this, since I won't know how to integrate z with respect to x. I do know this can easily be done by separating variables, however, we are basically told to solve it this way.
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Answers (2)

bamenyab4mxn
Answered 2022-05-17 Author has 16 answers
From y=zx, draw
y = z x + z .
Then
z x + z = 3 z z 2 3
which is separable:
z x = z 3 z z 2 3 = z 3 z 2 3 .
The equation can be seen as linear (and homogeneous) by swapping the roles of the dependent and independent variables and rewriting
d x d z + z 2 3 z 3 x = 0.
This doesn't ease the solution.
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llunallenaipg5r
Answered 2022-05-18 Author has 5 answers
Setting
y = u x
then we have
u x + u = 3 u u 2 3
simplifying we get
u x = u ( 3 u 2 3 + 1 )
or
u x = u 3 u 2 3
this equation is separble and it follows
u 2 3 u 3 d u = d x x
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