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
(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