I'm trying to solve an initial value problem. I know how to do the problem once integrated but solvi

gledanju0

gledanju0

Answered question

2022-06-11

I'm trying to solve an initial value problem. I know how to do the problem once integrated but solving the differential equation is where I'm finding trouble.
d y / d x = 3 + 2 y + 17 x 3
I'd thought that maybe squaring both sides will get rid of the square root but that doesn't work so I was hoping someone would point me in the right direction of how to go about seperating y and x .

Answer & Explanation

grcalia1

grcalia1

Beginner2022-06-12Added 23 answers

The differential equation
d y / d x = 3 + 2 y + 17 x 3
is not separable but a change of variable turns it into a separable equation.
Let
u = 2 y + 17 x 3
Then
d u / d x = 2 d y / d x + 17
Substitute in the original equation to get
d u / d x = 2 u + 23
Which is separable.
Solve for u and back substitute your solution to find y as a function of x
Gabriella Sellers

Gabriella Sellers

Beginner2022-06-13Added 4 answers

d y d x = 3 + 2 y + 17 x 3
Let 2 y + 17 x 3 = u 2 with
u 0 y = 17 2 + u u = 3 + u
u u = u + 23 2
x = u d u u + 23 2 = u 23 2 ln ( u + 23 2 ) + c
This gives x ( u ). The inverse function u ( x ) involves a special function:
u = 23 2 ( W ( X ) 1 ) where X = 2 23 exp ( 1 + 2 x c 23 )
W ( X ) is the Lambert's W function:
{ y ( x ) = 17 x + 3 2 + 1 2 ( 23 2 ( W ( X ) 1 ) ) 2 X = 2 23 exp ( 1 + 2 x c 23 )

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