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

ODE
\[ x y(x) \left (\sqrt {x^2-y(x)^2}+x\right ) y'(x)=x y(x)^2-\left (x^2-y(x)^2\right )^{3/2} \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
Homogeneous equation

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

\[\left \{\left \{y(x)\to -\sqrt {2} \sqrt {-\sqrt {2} \sqrt {-x^4 \left (c_1-\log (x)-1\right )}+\left (c_1-1\right ) x^2-x^2 \log (x)}\right \},\left \{y(x)\to \sqrt {2} \sqrt {-\sqrt {2} \sqrt {-x^4 \left (c_1-\log (x)-1\right )}+\left (c_1-1\right ) x^2-x^2 \log (x)}\right \},\left \{y(x)\to -\sqrt {2} \sqrt {\sqrt {2} \sqrt {-x^4 \left (c_1-\log (x)-1\right )}+\left (c_1-1\right ) x^2-x^2 \log (x)}\right \},\left \{y(x)\to \sqrt {2} \sqrt {\sqrt {2} \sqrt {-x^4 \left (c_1-\log (x)-1\right )}+\left (c_1-1\right ) x^2-x^2 \log (x)}\right \},\left \{y(x)\to -\sqrt {2} \sqrt {-\sqrt {2} \sqrt {x^4 \left (c_1-\log (x)+1\right )}-\left (c_1+1\right ) x^2+x^2 \log (x)}\right \},\left \{y(x)\to \sqrt {2} \sqrt {-\sqrt {2} \sqrt {x^4 \left (c_1-\log (x)+1\right )}-\left (c_1+1\right ) x^2+x^2 \log (x)}\right \},\left \{y(x)\to -\sqrt {2} \sqrt {\sqrt {2} \sqrt {x^4 \left (c_1-\log (x)+1\right )}-\left (c_1+1\right ) x^2+x^2 \log (x)}\right \},\left \{y(x)\to \sqrt {2} \sqrt {\sqrt {2} \sqrt {x^4 \left (c_1-\log (x)+1\right )}-\left (c_1+1\right ) x^2+x^2 \log (x)}\right \}\right \}\]

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

\[ \left \{ {\frac { \left (y \relax (x ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac {1}{x}\sqrt {{x}^{2}- \left (y \relax (x ) \right ) ^{2}}}+\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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