How tosolve this problem? (x+e^y)dy - dx = 0

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We are going to find `(dx)/(dy) = x + e^y` rewritiing as

`(dx)/(dy) - x = e^y`

This is a linear ode class 1.   We find `f(y)=-1` , `int f(y) dy = -y`

So we multiply both sides by `e^(-y)` and we get

`e^(-y)(dx)/(dy) - e^(-y)x = 1`

Now `(d)/(dy)xe^(-y) = e^(-y)(dx)/(dy)-e^(-y)x` which is the left side of our equation.  so

`(d(xe^(-y)))/(dy) = 1`

Integrating both sides with respect to y gives us

`xe^(-y) = y + C`

Solving for `x = e^y(y+C)` we get our solution.

We can see this is a solution because

`(dx)/(dy)=e^(y)(y+C)+e^y=x+e^y` which, multiplying both sides by dy and subtracting dx from both sides is what we started with.

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial