Show substitution leads to a separable differential equation. Consider the differential equation... y' = f(y/t) Show that the substitution v=v/t leads to a separable differential equation in v

Haiphongum 2022-09-15 Answered
Show substitution leads to a separable differential equation.
Consider the differential equation...
y = f ( y t )
Show that the substitution v = v t leads to a separable differential equation in v
Here's what I did.
v = y t
d v d t = d y d t 1 t 2
Sub into the orignal.
d v d t = f ( v ) 1 t 2
d v f ( v ) = 1 t 2 d t
This is where I get stuck. Is this the what the question is asking for, or am I forgetting to do something?
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Answers (1)

Claire Larson
Answered 2022-09-16 Author has 10 answers
Note that if you substitute v = y t y = v t. Note that v is assumed to be a function of t. Using the product rule, we obtain the derivative y′:
(1) y = v ( t ) t y = d v d t t + v
Substituting into y = f ( y t ) :
d v d t t + v = f ( v )
Now, try to separate the variables. i.e, put only v and dv terms on one side of the equality and t and dt terms on the other side, then you are done.
Feel free to ask on the comments if you have any related doubts or questions.

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