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

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.