अवकल समीकरण `(dy)/(dx)+(1+y^(3))/(xy^(2)(1+x^(2)))=0` को हल कीजिये।

Explanation

`(dy)/(dx)+(1+y^(3))/(xy^(2)(1+x^(2)))=0`
`implies(y^(2)dy)/(1+y^(2))+(dx)/(x(1+x^(2)))=0`
`implies(y^(2)dy)/(1+y^(3))+(xdx)/(x^(2)(1+x^(2)))=0`
समाकलन से,
`int(y^(2)dy)/(1+y^(2))+int(xdx)/(x^(2)(1+x^(2)))=c" "......(1)`
माना `1+y^(3)=t_(1)implies3y^(2)dy=dt`
`impliesy^(2)dy=(1)/(3)dt_(1)`
तथा `x^(2)=t_(1)impliesxdx=(1)/(2)dt_(2)`
अतः समीकरण (1) से,
`(1)/(3)int(dt_(1))/(t_(1))+(1)/(2)int(dt_(2))/(t_(2)(1+t_(2)))=c`
`implies(1)/(3)int(dt_(1))/(t_(1))+(1)/(2)int((1)/(t_(2))-(1)/(1+t_(2)))dt_(2)=c`
(द्वितीय फलन में आंशिक भिन्न की क्रिया की गई है।)
`implies(1)/(3)logt_(1)+(1)/(2)[logt_(2)-log(1+t_(2))]=c`
`implies(1)/(3)logt_(1)+(1)/(2)[log((t_(2))/(1+t_(2)))]=c`
`implies2logt_(1)+3log((t_(2))/(1+t_(2)))=6c" "` (6 से गुणा करने पर)
`implieslog_(1)^(2)+log((t_(1)/(1+t_(2)))=logk" "` (माना)
`implieslog((t_(1)^(2)t_(2)^(3))/((1+t_(2))^(3)))=logk`
`impliest_(1)^(2)t_(2)^(3)=k(1+t_(2))^(3)`
`implies(1+y^(3))^(2)x^(6)=k(1+x^(2))^(3)`