I'm taking an elementary differential equations class and we got our first exam back and...

Sovardipk

Sovardipk

Answered

2022-07-06

I'm taking an elementary differential equations class and we got our first exam back and I don't understand why I was wrong on one of the questions. I got a 96, and the professor said there were quite a few d's and f's, so I didn't want to quibble about this.
Anyway, the problem was d y d x = 1 x + y + 2 and I chose the sub u = x + y + 2, with d y d x = d y d u d u d x = d y d u ( 1 + 1 u ) and then plugged in the expression for d y d x , separated the equation and solved for y in terms of u and then substituted back to get y = ln ( x + y + 3 ) + c . The professor took issue with my expression for d y d x , saying that I couldn't differentiate y w.r.t. u if y is part of u. Is this correct?

Answer & Explanation

Yair Boyle

Yair Boyle

Expert

2022-07-07Added 10 answers

Your solution seems fine, although it is an implicit solution. I think an easier way to solve the DE
d y d x = 1 x + y + 2
is to arrange as
d x d y = x + y + 2.
This is a first-order linear equation for x(y). I get
x ( y ) = C e y y 3.
It's possible to solve for y using the Lambert W function, but it's rather messy and doesn't provide as much insight.

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