Find the special solution of dy / DX = x + XY ^ / y + YX ^ satisfying the initial condition y (x = 0) = 2

Find the special solution of dy / DX = x + XY ^ / y + YX ^ satisfying the initial condition y (x = 0) = 2

dy/dx = (x+xy²)/(y+yx²)
(y+yx²)dy = (x+xy²)dx
ydy + yx²dy - xdx - xy²dx = 0
ydy - xdx + 1/2 * (x²dy² - y²dx²) = 0
dy² - dx² + x²dy² - y²dx² = 0
(1+x²)dy² = (1+y²)dx²
dy²/(1+y²) = dx²/(1+x²)
ln(1+y²) = ln(1+x²) + C
When x = 0, y = 2
be
ln5 = C
The special solution is
ln(1+y²) = ln(1+x²) + ln5
Namely
y = √(4+5x²)

Elden Ring Premiere

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