अवकल समीकरण `(x + y)dy + (x-y)dx = 0` को हल कीजिए |

अवकल समीकरण `(x + y)dy + (x-y)dx = 0` को ह�

1.	अवकल समीकरण `(x + y)dy + (x-y)dx = 0` को हल कीजिए \|
Answer» `because " " (x+y) dy + (x -y) dx = 0` `(x+y) dy = (y-x) dx` `(dy)/(dx) = (y-x)/(x+Y)` यह एक समघाती अवकल समीकरण है \| यदि `y = vx,` तब `" " (dy)/(dx) = v + x (dv)/(dx)` `therefore " " v +x (dv)/(dx) = (vx -x)/(x+vx)` `x(dv)/(dx) = (v-1)/(v+1) -v` `= (v-1-v^(2)-v)/(v+1) = (-(v^(2) + 1))/(v+1)` `therefore " " (v+1)/(v^(2) + 1) dv + (edv)/(x) =0` दोनों पक्षों का समाकलन करने पर, `f""(v)/(v^(2)+1)dv+f""(1)/(v^(2)+1)dv+f""(dv)/(x) =c` `rArr (1)/(2)f""(2v)/(v^(2)+1)dv+f""(dv)/(v^(2)+1)+f""(dx)/(x) =c` `rArr (1)/(2)log(v^(2)+1)+tan^(-1) + logx =c` ` rArr (1)/(2)log (v^(2) + 1) + log x + tan^(-1)v =c ` `rArr (log(v^(2)+1)+21logx)/(2) + tan^(-1) v =c ` `rArr (log(v^(2) + 1) + log x ^(2))/(2) + tan^(-1) v =c ` `rArr (1)/(2) log (v^(2) + 1)x^(2) + tan^(-1) v =c ` `rArr (1)/(2) log(y^(2) + x^(2)) + tan^(-1)""(y)/(x) =c`.