Question

(1+x^2+y^2+x^2y^2)^1/2+xydy/dx

Answer 1

Final Answer :
$\sqrt{(1 + {x}^{2} )} + \frac{1}{2} ln( \frac{ \sqrt{1 + {x}^{2} } - 1}{ \sqrt{1 + {x}^{2} } + 1 } ) \\ = - \sqrt{1 + {y}^{2} } + c \: where \: c \: \: is \: \\ arbitrary \: constant \: .$


It will be meaningless Question if you are not equating Given ODE = 0

So ,
Edit: ODE (given) = 0


Now, Solving this ODE,


Question Edit:

Concepts :
1) Substitution Method of Integrals.

2) Integration of
$\frac{1}{ {t}^{2} - 1 } dt \: = \frac{1}{2} ln( \frac{t - 1}{t + 1} ) + c \: \\ where \: c \: is \: arbitrary \: constant \: .$

Steps involved :

1) Factorise the Square root expression.

2) Put 1+x^2=t^2

3) Solve in terms of t and y and then use
above integral indentity whose proof can be
derived by substituting t = sin a .


For Calculation Process :
See Pic attached to it.



Hope, you understand my answer and it may helps you.