4.16.37 \(y'(x)^2-x y'(x)-y(x)=0\)

ODE
\[ y'(x)^2-x y'(x)-y(x)=0 \] ODE Classification

[[_1st_order, _with_linear_symmetries], _dAlembert]

Book solution method
No Missing Variables ODE, Solve for \(y\)

Mathematica
cpu = 0.28834 (sec), leaf count = 1493

\[\left \{\left \{y(x)\to \frac {x^2}{2}+\frac {\left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{4 \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {1}{4} \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{8 \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {1}{8} i \left (i+\sqrt {3}\right ) \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {i \left (i+\sqrt {3}\right ) \left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{8 \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}-\frac {1}{8} i \left (-i+\sqrt {3}\right ) \sqrt [3]{-x^6+20 \cosh \left (3 c_1\right ) x^3+20 \sinh \left (3 c_1\right ) x^3+8 \cosh \left (6 c_1\right )+8 \sinh \left (6 c_1\right )+8 \sqrt {-\left (\left (x^3-1\right ) \cosh \left (\frac {3 c_1}{2}\right )-\left (x^3+1\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {\left (x^3-2 \cosh \left (3 c_1\right )-2 \sinh \left (3 c_1\right )\right ) x}{2\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {\sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{4 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {x^2}{2}-\frac {i \left (-i+\sqrt {3}\right ) \left (x^3-2 \cosh \left (3 c_1\right )-2 \sinh \left (3 c_1\right )\right ) x}{4\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}+\frac {i \left (i+\sqrt {3}\right ) \sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{8 \sqrt [3]{2}}\right \},\left \{y(x)\to \frac {x^2}{2}+\frac {i \left (i+\sqrt {3}\right ) \left (x^3-2 \cosh \left (3 c_1\right )-2 \sinh \left (3 c_1\right )\right ) x}{4\ 2^{2/3} \sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}-\frac {i \left (-i+\sqrt {3}\right ) \sqrt [3]{-2 x^6-10 \cosh \left (3 c_1\right ) x^3-10 \sinh \left (3 c_1\right ) x^3+\cosh \left (6 c_1\right )+\sinh \left (6 c_1\right )+\sqrt {\left (\left (4 x^3+1\right ) \cosh \left (\frac {3 c_1}{2}\right )+\left (1-4 x^3\right ) \sinh \left (\frac {3 c_1}{2}\right )\right ){}^3 \left (\cosh \left (\frac {15 c_1}{2}\right )+\sinh \left (\frac {15 c_1}{2}\right )\right )}}}{8 \sqrt [3]{2}}\right \}\right \}\]

Maple
cpu = 0.016 (sec), leaf count = 31

\[ \left \{ [x \left ({\it \_T} \right ) ={1 \left ({\frac {2}{3}{{\it \_T}}^{{\frac {3}{2}}}}+{\it \_C1} \right ) {\frac {1}{\sqrt {{\it \_T}}}}},y \left ({\it \_T} \right ) ={\frac {{{\it \_T}}^{2}}{3}}-\sqrt {{\it \_T}}{\it \_C1}] \right \} \] Mathematica raw input

DSolve[-y[x] - x*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> x^2/2 + (x*(x^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]]))/(4*(-x^6 + 20*x^3*
Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(
((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] 
+ Sinh[(15*C[1])/2]))])^(1/3)) + (-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 
20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (
1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3)/4}
, {y[x] -> x^2/2 - ((I/8)*(-I + Sqrt[3])*x*(x^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]
]))/(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[
6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*
(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3) + (I/8)*(I + Sqrt[3])*(-x^6 + 2
0*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*S
qrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1
])/2] + Sinh[(15*C[1])/2]))])^(1/3)}, {y[x] -> x^2/2 + ((I/8)*(I + Sqrt[3])*x*(x
^3 + 8*Cosh[3*C[1]] + 8*Sinh[3*C[1]]))/(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[
1]] + 20*x^3*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/
2] - (1 + x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1
/3) - (I/8)*(-I + Sqrt[3])*(-x^6 + 20*x^3*Cosh[3*C[1]] + 8*Cosh[6*C[1]] + 20*x^3
*Sinh[3*C[1]] + 8*Sinh[6*C[1]] + 8*Sqrt[-(((-1 + x^3)*Cosh[(3*C[1])/2] - (1 + x^
3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]))])^(1/3)}, {y[x] 
-> x^2/2 + (x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C[1]]))/(2*2^(2/3)*(-2*x^6 - 10*x
^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 +
 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + 
Sinh[(15*C[1])/2])])^(1/3)) + (-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*
x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^
3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3)/(4*2^(1/3
))}, {y[x] -> x^2/2 - ((I/4)*(-I + Sqrt[3])*x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C
[1]]))/(2^(2/3)*(-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1
]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1
])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3)) + ((I/8)*(I + Sqrt[3])
*(-2*x^6 - 10*x^3*Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1
]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[
(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3))/2^(1/3)}, {y[x] -> x^2/2 + ((I/4)*(I 
+ Sqrt[3])*x*(x^3 - 2*Cosh[3*C[1]] - 2*Sinh[3*C[1]]))/(2^(2/3)*(-2*x^6 - 10*x^3*
Cosh[3*C[1]] + Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*
x^3)*Cosh[(3*C[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sin
h[(15*C[1])/2])])^(1/3)) - ((I/8)*(-I + Sqrt[3])*(-2*x^6 - 10*x^3*Cosh[3*C[1]] +
 Cosh[6*C[1]] - 10*x^3*Sinh[3*C[1]] + Sinh[6*C[1]] + Sqrt[((1 + 4*x^3)*Cosh[(3*C
[1])/2] + (1 - 4*x^3)*Sinh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2]
)])^(1/3))/2^(1/3)}}

Maple raw input

dsolve(diff(y(x),x)^2-x*diff(y(x),x)-y(x) = 0, y(x),'implicit')

Maple raw output

[x(_T) = 1/_T^(1/2)*(2/3*_T^(3/2)+_C1), y(_T) = 1/3*_T^2-_T^(1/2)*_C1]