ODE
\[ y'(x)^3+x y'(x)^2-y(x)=0 \] ODE Classification
[_dAlembert]
Book solution method
No Missing Variables ODE, Solve for \(y\)
Mathematica ✓
cpu = 53.0238 (sec), leaf count = 1496
\[\left \{\left \{y(x)\to \frac {-16 x^4+8 \left (\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}-12\right ) x^3-4 \left (\left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}-9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right ) x^2+6 \left (72 c_1+2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+4 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}\right ) x+3 \left (4 c_1 \left (2 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+9 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+54\right )+9 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}+12 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+2 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )} \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+27 \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+81\right )}{24 \left (-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27\right ){}^{2/3}}\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {1}{48} \left (-\frac {i \left (-i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x+i \left (i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right ){}^2+\frac {1}{3} x \left (-\frac {i \left (-i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x+i \left (i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right )-2 x+2 c_1\right )\right \},\left \{y(x)\to \frac {1}{2} \left (\frac {1}{48} \left (\frac {i \left (i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x-i \left (-i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right ){}^2+\frac {1}{3} x \left (\frac {i \left (i+\sqrt {3}\right ) (2 x+3)^2}{\sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}}-4 x-i \left (-i+\sqrt {3}\right ) \sqrt [3]{-8 x^3-36 x^2-54 x+108 c_1+6 \sqrt {6} \sqrt {\left (2 c_1+1\right ) \left (-4 x^3-18 x^2-27 x+27 c_1\right )}+27}+6\right )-2 x+2 c_1\right )\right \}\right \}\]
Maple ✓
cpu = 0.021 (sec), leaf count = 50
\[ \left \{ y \left ( x \right ) =0,[x \left ( {\it \_T} \right ) ={\frac {1}{ \left ( {\it \_T}-1 \right ) ^{2}} \left ( -{{\it \_T}}^{3}+{\frac {3\,{{\it \_T}}^{2}}{2}}+{\it \_C1} \right ) },y \left ( {\it \_T} \right ) =-{\frac {{{\it \_T}}^{2} \left ( {{\it \_T}}^{2}-2\,{\it \_C1}-2\,{\it \_T} \right ) }{2\, \left ( {\it \_T}-1 \right ) ^{2}}}] \right \} \] Mathematica raw input
DSolve[-y[x] + x*y'[x]^2 + y'[x]^3 == 0,y[x],x]
Mathematica raw output
{{y[x] -> (-16*x^4 + 8*x^3*(-12 + (27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3)) - 4*x^2*(54 - 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + (27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3)) + 6*x*(72*C[1] + 4*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])] + 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 2*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3)) + 3*(81 + 12*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])] + 27*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 2*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])]*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3) + 4*C[1]*(54 + 9*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + 2*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3))))/(24*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(2/3))}, {y[x] -> (-2*x + 2*C[1] + (x*(6 - 4*x - (I*(-I + Sqrt[3])*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + I*(I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3)))/3 + (6 - 4*x - (I*(-I + Sqrt[3])*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) + I*(I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3))^2/48)/2}, {y[x] -> (-2*x + 2*C[1] + (x*(6 - 4*x + (I*(I + Sqrt[3])*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) - I*(-I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3)))/3 + (6 - 4*x + (I*(I + Sqrt[3])*(3 + 2*x)^2)/(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3) - I*(-I + Sqrt[3])*(27 - 54*x - 36*x^2 - 8*x^3 + 108*C[1] + 6*Sqrt[6]*Sqrt[(1 + 2*C[1])*(-27*x - 18*x^2 - 4*x^3 + 27*C[1])])^(1/3))^2/48)/2}}
Maple raw input
dsolve(diff(y(x),x)^3+x*diff(y(x),x)^2-y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = 0, [x(_T) = 1/(_T-1)^2*(-_T^3+3/2*_T^2+_C1), y(_T) = -1/2*_T^2*(_T^2-2*_C1-2*_T)/(_T-1)^2]