(1+x2+y2+x2y2) 1/2 + xy dy/dx =0 how to solve dis diffrential equation! ! please ans asap!! :) - Maths - Differential Equations

Question

(1+x2+y2+x2y2)
1/2 + xy dy/dx =0 how to solve dis diffrential
equation!?! please ans asap!! :)

Ajanta Trivedi · Accepted Answer

the given differential equation is:

$(1 + x^{2} + y^{2} + x^{2} y^{2})^{1 / 2} + x y \cdot \frac{d y}{d x} = 0 [(1 + x^{2}) + y^{2} (1 + x^{2})]^{1 / 2} + x y \cdot \frac{d y}{d x} = 0 (1 + x^{2})^{1 / 2} \cdot (1 + y^{2})^{1 / 2} + x y \cdot \frac{d y}{d x} = 0$

$(1 + x^{2})^{1 / 2} \cdot (1 + y^{2})^{1 / 2} = - x y \cdot \frac{d y}{d x} \frac{\sqrt{1 + x^{2}}}{x} d x = - \frac{y}{\sqrt{1 + y^{2}}} d y$ ..................(1)

now integrating (1):

$\int \frac{\sqrt{1 + x^{2}}}{x} ⅆ x = - \int \frac{y}{\sqrt{1 + y^{2}}} ⅆ y + C$ ...........(2)where C is an integration constant

let $I_{1} = \int \frac{\sqrt{1 + x^{2}}}{x} ⅆ x = \int \frac{\sqrt{1 + x^{2}}}{x^{2}} \cdot x ⅆ x$ let $1 + x^{2} = t^{2} \Rightarrow 2 x d x = 2 t d t i . e . x d x = t d t$

$I_{1} = \int \frac{t}{t^{2} - 1} \cdot t ⅆ t = \int \frac{t^{2} - 1 + 1}{t^{2} - 1} ⅆ t = \int [1 + \frac{1}{t^{2} - 1}] ⅆ t I_{1} = \int [1 + \frac{1}{2 (t - 1)} - \frac{1}{2 (t + 1)}] ⅆ t$ [by partial fraction]

now integrating:

$I_{1} = t + \frac{1}{2} \log | t - 1 | - \frac{1}{2} \log | t + 1 | I_{1} = \sqrt{1 + x^{2}} + \frac{1}{2} \log | \sqrt{1 + x^{2}} - 1 | - \frac{1}{2} \log | \sqrt{1 + x^{2}} + 1 | I_{1} = \sqrt{1 + x^{2}} + \frac{1}{2} \log | \frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2}} + 1} |$ ................(3)

let $I_{2} = \int \frac{y}{\sqrt{1 + y^{2}}} ⅆ y$ let $1 + y^{2} = u \Rightarrow 2 y d y = d u$

$I_{2} = \frac{1}{2} \cdot \int \frac{1}{\sqrt{u}} ⅆ u = \frac{1}{2} \cdot 2 \sqrt{u} I_{2} = \sqrt{1 + y^{2}}$ .................(4)

from (2), (3) and (4): we have,

$\sqrt{1 + x^{2}} + \frac{1}{2} \log | \frac{\sqrt{1 + x^{2}} - 1}{\sqrt{1 + x^{2}} + 1} | + \sqrt{1 + y^{2}} = C$

which is the required result.

hope this helps you.

(1+x2+y2+x2y2) 1/2 + xy dy/dx =0 how to solve dis diffrential equation!?! ..please ans asap!! :)

(1+x²+y²+x²y²) ^1/2 + xy dy/dx =0 how to solve dis diffrential equation!?! ..please ans asap!! :)