हल कीजिए : `(dy)/(dx) = (x^(2) + 5xy + 4y^(2))/(x^(2))`

1.	हल कीजिए : `(dy)/(dx) = (x^(2) + 5xy + 4y^(2))/(x^(2))`
Answer» `(dy)/(dx) = (x^(2) + 5xy + 4y^(2))/(x^(2))" "…(i)` यह एक समघाती अवकल समीकरण है, अतः यदि `y = vx" "…(ii)` `(dy)/(dx) = v + x (dv)/(dx)` समीकरण, (i) से, `" " v + x (dv)/(dx) = (x^(2) + 5vx^(2) + 4v^(2)x^(2))/(x^(2))` `" " = 1 + 5v + 4v^(2)` `x(dv)/(dx) = 1 + 4v + 4v^(2) = (2v + 1)^(2)` या `" " (dv)/((2v + 1)^(2)) = (dx)/(x)` दोनों पक्षों में समाकलन करने पर, `f""(dv)/((2v + 1)^(2)) = f""(dx)/(x) + c` `(1)/(2""(2v+1)) = log x + c ` समीकरण (ii) से, `(1)/(2(2(y)/(x)+1))=logx+c` `(-x)/(2(x+2y))=log x + c`

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