The possible solution of the differential equation y(5x2−2y2)dx=x(5y2−3x3)dyisxayb(xc−yd)=k (where k is arbitrary constant) then value of a+b+c+d is

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Solution

$(5 x^{2} y - 2 y^{3}) d x = (5 x y^{2} - 3 x^{4}) d y$

$5 x^{3} y d x + 3 x^{4} d y - 2 y^{3} d x - 5 x y^{2} d y = 0$

multiplying equation by $x y^{2}$

$5 x^{4} y^{3} d x + 3 x^{4} y^{2} d y, - 2 x y^{5} d x - 5 x^{2} y^{4} d y = 0$

$d (x^{5} y^{3}) - d (x^{2} y^{5}) = 0$

Integrating both sides

$x^{5} y^{3} - x^{2} y^{5} = k$

$x^{2} y^{3} (x^{3} - y^{2}) = k$

$a = 2, b = 3, c = 3, d = 2$

$a + b + c + d = 10$

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