Solve the following differential equations: √(1 + x^2 + y^2 + x^2y^2) + xy(dy/dx) = 0

Question

Solve the following differential equations:

\(\sqrt{1+x^2+y^2+x^2y^2}+xy\frac{dy}{dx}=0\)

Answer 1

Re - writing the equation as

Now separating variable x on one side and variable y on another side, we have

Multiplying and dividing the numerator of LHS by x

Assuming 1 + y² = t² and 1 + x² = v²

Differentiating we get,

ydy = tdt

xdx = vdv

substituting these values in above differential equation

Integrating both sides

Adding 1 and subtracting 1 to the numerator of RHS

Substituting t and v in above equation