xy(x)( ∘ _______ x2 −y(x)2 + x) y′(x) = xy(x)2 −( x2 −y(x)2) 3∕2

4.15.36 $x y (x) (\sqrt{x^{2} - y (x)^{2}} + x) y^{'} (x) = x y (x)^{2} - {(x^{2} - y (x)^{2})}^{3 / 2}$

ODE
$x y (x) (\sqrt{x^{2} - y (x)^{2}} + x) y^{'} (x) = x y (x)^{2} - {(x^{2} - y (x)^{2})}^{3 / 2}$ ODE Classiﬁcation

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
Homogeneous equation

Mathematica ✓
cpu = 1.22711 (sec), leaf count = 429

${{y (x) \to - \sqrt{2} \sqrt{- \sqrt{2} \sqrt{- x^{4} (c_{1} - \log (x) - 1)} + (c_{1} - 1) x^{2} - x^{2} \log (x)}}, {y (x) \to \sqrt{2} \sqrt{- \sqrt{2} \sqrt{- x^{4} (c_{1} - \log (x) - 1)} + (c_{1} - 1) x^{2} - x^{2} \log (x)}}, {y (x) \to - \sqrt{2} \sqrt{\sqrt{2} \sqrt{- x^{4} (c_{1} - \log (x) - 1)} + (c_{1} - 1) x^{2} - x^{2} \log (x)}}, {y (x) \to \sqrt{2} \sqrt{\sqrt{2} \sqrt{- x^{4} (c_{1} - \log (x) - 1)} + (c_{1} - 1) x^{2} - x^{2} \log (x)}}, {y (x) \to - \sqrt{2} \sqrt{- \sqrt{2} \sqrt{x^{4} (c_{1} - \log (x) + 1)} - (c_{1} + 1) x^{2} + x^{2} \log (x)}}, {y (x) \to \sqrt{2} \sqrt{- \sqrt{2} \sqrt{x^{4} (c_{1} - \log (x) + 1)} - (c_{1} + 1) x^{2} + x^{2} \log (x)}}, {y (x) \to - \sqrt{2} \sqrt{\sqrt{2} \sqrt{x^{4} (c_{1} - \log (x) + 1)} - (c_{1} + 1) x^{2} + x^{2} \log (x)}}, {y (x) \to \sqrt{2} \sqrt{\sqrt{2} \sqrt{x^{4} (c_{1} - \log (x) + 1)} - (c_{1} + 1) x^{2} + x^{2} \log (x)}}}$

Maple ✓
cpu = 0.061 (sec), leaf count = 35

${\frac{{(y (x))}^{2}}{2 x^{2}} - \frac{1}{x} \sqrt{x^{2} - {(y (x))}^{2}} + \ln (x) -_C 1 = 0}$ Mathematica raw input

DSolve[x*y[x]*(x + Sqrt[x^2 - y[x]^2])*y'[x] == x*y[x]^2 - (x^2 - y[x]^2)^(3/2),y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[2]*Sqrt[x^2*(-1 + C[1]) - Sqrt[2]*Sqrt[-(x^4*(-1 + C[1] - Log[x
]))] - x^2*Log[x]])}, {y[x] -> Sqrt[2]*Sqrt[x^2*(-1 + C[1]) - Sqrt[2]*Sqrt[-(x^4
*(-1 + C[1] - Log[x]))] - x^2*Log[x]]}, {y[x] -> -(Sqrt[2]*Sqrt[x^2*(-1 + C[1])
+ Sqrt[2]*Sqrt[-(x^4*(-1 + C[1] - Log[x]))] - x^2*Log[x]])}, {y[x] -> Sqrt[2]*Sq
rt[x^2*(-1 + C[1]) + Sqrt[2]*Sqrt[-(x^4*(-1 + C[1] - Log[x]))] - x^2*Log[x]]}, {
y[x] -> -(Sqrt[2]*Sqrt[-(x^2*(1 + C[1])) - Sqrt[2]*Sqrt[x^4*(1 + C[1] - Log[x])]
+ x^2*Log[x]])}, {y[x] -> Sqrt[2]*Sqrt[-(x^2*(1 + C[1])) - Sqrt[2]*Sqrt[x^4*(1
+ C[1] - Log[x])] + x^2*Log[x]]}, {y[x] -> -(Sqrt[2]*Sqrt[-(x^2*(1 + C[1])) + Sq
rt[2]*Sqrt[x^4*(1 + C[1] - Log[x])] + x^2*Log[x]])}, {y[x] -> Sqrt[2]*Sqrt[-(x^2
*(1 + C[1])) + Sqrt[2]*Sqrt[x^4*(1 + C[1] - Log[x])] + x^2*Log[x]]}}

Maple raw input

dsolve(x*y(x)*(x+(x^2-y(x)^2)^(1/2))*diff(y(x),x) = x*y(x)^2-(x^2-y(x)^2)^(3/2), y(x),'implicit')

Maple raw output

1/2/x^2*y(x)^2-1/x*(x^2-y(x)^2)^(1/2)+ln(x)-_C1 = 0

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4.15.36 xy(x)(x2−y(x)2+x)y′(x)=xy(x)2−(x2−y(x)2)3/2

4.15.36 $x y (x) (\sqrt{x^{2} - y (x)^{2}} + x) y^{'} (x) = x y (x)^{2} - {(x^{2} - y (x)^{2})}^{3 / 2}$