Solve the differential equation dy∕dx = (y²+4)/(x²+16), y(4)=1.

Expert Answers

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

Hello!

This differential equation is a separable one, it is possible to separate `y` from `x.` For this, simply divide both sides by `(y^2+4):`

`(y')/(y^2+4) = 1/(x^2+16).`

`y` is at the left side only, `x` is at the right side only. Moreover, both sides are integrable in elementary functions:

`1/2 arctan(y/2) = 1/4 arctan(x/4)+C,`

or  `arctan(y/2) = 1/2 arctan(x/4)+C.`  (1)

 

Now use the given boundary condition, `y(4)=1,` to find `C:`

`arctan(1/2) = 1/2 arctan(1) + C,` or

`C =arctan(1/2) - 1/2 arctan(1) = arctan(1/2) - pi/8.`

 

If we take `tan` of the both sides of (1), we obtain

`y(x)=2tan(1/2 arctan(x/4)+arctan(1/2)-pi/8).`

This is the only solution.

Approved by eNotes Editorial Team

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