What is the general solution of the differential equation dy/dx=(y+2x)^2+2 ?

1 Answer
Sep 27, 2017

y = 2tan(2x + A) - 2x

Explanation:

We have:

dy/dx=(y+2x)^2+2 ..... [A]

We can perform a substitution, Let

v = y + 2x

Then if we differentiate, we get:

(dv)/dx = dy/dx + 2 => dy/dx = (dv)/dx - 2

Substituting into the original DE A, we get:

(dv)/dx - 2 = v^2+2
:. (dv)/dx = v^2+4

This is now a First Order separable DE, so we can rearrange and seperate the variables, giving:

int \ 1/(v^2+4) \ dv = int \ dx

Which we can integrate, using standard integrals, to get:

1/2arctan (v/2) = x + C

And, restoring the substitution:

1/2arctan ((y + 2x)/2) = x + C
arctan ((y + 2x)/2) = 2x + 2C
:. (y + 2x)/2 = tan(2x + A)
:. y+2x = 2tan(2x + A)
:. y = 2tan(2x + A) - 2x