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 = 0.539481 (sec), leaf count = 1445
\[\left \{\left \{y(x)\to \frac {1}{36} \left (-9 x^2-\frac {9 \left (x^3+8 \cosh (3 c_1)+8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1)+8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1)+8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6-20 \cosh (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3-20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1)-8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1)-8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1)-8 \sinh (3 c_1)\right ) x}{\sqrt [3]{x^6+20 \cosh (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 (3 c_1) x^3+20 \sinh (3 c_1) x^3-8 \cosh (6 c_1)-8 \sinh (6 c_1)+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 = 0.044 (sec), leaf count = 690
\[\left [y \left (x \right ) = \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right )^{2}+2 x \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}+\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}\right ), y \left (x \right ) = \left (-\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}-\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ), y \left (x \right ) = \left (-\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2 x \left (-\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{4}-\frac {x^{2}}{4 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}-\frac {x}{2}+\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}{2}-\frac {x^{2}}{2 \left (6 \textit {\_C1} -x^{3}+2 \sqrt {-3 \textit {\_C1} \,x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right )\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))/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)*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*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}, {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))/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)*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*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))
Maple raw output
[y(x) = (1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)+1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x)^2+2*x*(1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)+1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x), y(x) = (-1/4*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/4*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x-1/2*I*3^(1/2)*(1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)))^2+2*x*(-1/4*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/4*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x-1/2*I*3^(1/2)*(1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3))), y(x) = (-1/4*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/4*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x+1/2*I*3^(1/2)*(1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)))^2+2*x*(-1/4*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/4*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x+1/2*I*3^(1/2)*(1/2*(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)-1/2*x^2/(6*_C1-x^3+2*(-3*_C1*x^3+9*_C1^2)^(1/2))^(1/3)))]