4.16.45 \(y'(x)^2+2 x y'(x)-y(x)=0\)

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

[[_1st_order, _with_linear_symmetries], _dAlembert]

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

Mathematica
cpu = 0.404325 (sec), leaf count = 1445

\[\left \{\left \{y(x)\to \frac {1}{36} \left (-9 x^2-\frac {9 \left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}-9 \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 )\right \},\left \{y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1+i \sqrt {3}\right ) \left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}+9 \left (1-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 )\right \},\left \{y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1-i \sqrt {3}\right ) \left (x^3+8 \cosh \left (3 c_1\right )+8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}+9 \left (1+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 )\right \},\left \{y(x)\to \frac {1}{36} \left (-9 x^2-\frac {9 \left (x^3-8 \cosh \left (3 c_1\right )-8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}-9 \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 )\right \},\left \{y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1+i \sqrt {3}\right ) \left (x^3-8 \cosh \left (3 c_1\right )-8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}+9 \left (1-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 )\right \},\left \{y(x)\to \frac {1}{72} \left (-18 x^2+\frac {9 \left (1-i \sqrt {3}\right ) \left (x^3-8 \cosh \left (3 c_1\right )-8 \sinh \left (3 c_1\right )\right ) x}{\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 )}}}+9 \left (1+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 )\right \}\right \}\]

Maple
cpu = 0.014 (sec), leaf count = 33

\[ \left \{ [x \left ({\it \_T} \right ) ={\frac {1}{{{\it \_T}}^{2}} \left (-{\frac {2\,{{\it \_T}}^{3}}{3}}+{\it \_C1} \right ) },y \left ({\it \_T} \right ) ={\frac {-{{\it \_T}}^{3}+6\,{\it \_C1}}{3\,{\it \_T}}}] \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-9*x^2 - (9*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) - 9*(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))/3
6}, {y[x] -> (-18*x^2 + (9*(1 + 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) + 9*(1 - 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*Sqr
t[-(((-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))/72}, {y[x] -> (-18*x^2 + (9*(1 - 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) + 9*(1 + I*Sqrt[3])*(x^6 - 20*x^3*Cosh[3*C[1]] - 8*Cosh[6*C[1]] - 20*x^3*S
inh[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))/72}, {y[x
] -> (-9*x^2 - (9*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) - 9*(x^6 + 20*x^3*Cosh[3*C[1]] - 8*Cosh[6*C[1]] + 20*x^3*S
inh[3*C[1]] - 8*Sinh[6*C[1]] + 8*Sqrt[((1 + x^3)*Cosh[(3*C[1])/2] - (-1 + x^3)*S
inh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3))/36}, {y[x] -
> (-18*x^2 + (9*(1 + 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) + 9*(1 - 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)*C
osh[(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))/72}, {y[x] -> (-18*x^2 + (9*(1 - 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*Si
nh[3*C[1]] - 8*Sinh[6*C[1]] + 8*Sqrt[((1 + x^3)*Cosh[(3*C[1])/2] - (-1 + x^3)*Si
nh[(3*C[1])/2])^3*(Cosh[(15*C[1])/2] + Sinh[(15*C[1])/2])])^(1/3) + 9*(1 + I*Sqr
t[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))/72}}

Maple raw input

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

Maple raw output

[x(_T) = 1/_T^2*(-2/3*_T^3+_C1), y(_T) = 1/3*(-_T^3+6*_C1)/_T]