ODE
\[ \left (x-y'(x)^2\right ) y''(x)=x^2-y'(x) \] ODE Classification
[[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_poly_yn]]
Book solution method
TO DO
Mathematica ✓
cpu = 3.95572 (sec), leaf count = 394
\[\left \{\left \{y(x)\to \int _1^x\left (-\frac {\sqrt [3]{2} K[1]}{\sqrt [3]{K[1]^3+3 c_1+\sqrt {K[1]^6+6 c_1 K[1]^3-4 K[1]^3+9 c_1{}^2}}}-\frac {\sqrt [3]{K[1]^3+3 c_1+\sqrt {K[1]^6+6 c_1 K[1]^3-4 K[1]^3+9 c_1{}^2}}}{\sqrt [3]{2}}\right )dK[1]+c_2\right \},\left \{y(x)\to \int _1^x\frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) K[2]+2^{2/3} \left (1-i \sqrt {3}\right ) \left (K[2]^3+3 c_1+\sqrt {K[2]^6+6 c_1 K[2]^3-4 K[2]^3+9 c_1{}^2}\right ){}^{2/3}}{4 \sqrt [3]{K[2]^3+3 c_1+\sqrt {K[2]^6+6 c_1 K[2]^3-4 K[2]^3+9 c_1{}^2}}}dK[2]+c_2\right \},\left \{y(x)\to \int _1^x\frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) K[3]+2^{2/3} \left (1+i \sqrt {3}\right ) \left (K[3]^3+3 c_1+\sqrt {K[3]^6+6 c_1 K[3]^3-4 K[3]^3+9 c_1{}^2}\right ){}^{2/3}}{4 \sqrt [3]{K[3]^3+3 c_1+\sqrt {K[3]^6+6 c_1 K[3]^3-4 K[3]^3+9 c_1{}^2}}}dK[3]+c_2\right \}\right \}\]
Maple ✓
cpu = 0.796 (sec), leaf count = 348
\[\left [y \left (x \right ) = \int \frac {\left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {2}{3}}+4 x}{2 \left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}d x +\textit {\_C2}, y \left (x \right ) = \int -\frac {i \sqrt {3}\, \left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x +\left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {2}{3}}+4 x}{4 \left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}d x +\textit {\_C2}, y \left (x \right ) = \int \frac {i \sqrt {3}\, \left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, x -\left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {2}{3}}-4 x}{4 \left (-4 x^{3}+12 \textit {\_C1} +4 \sqrt {x^{6}-6 \textit {\_C1} \,x^{3}-4 x^{3}+9 \textit {\_C1}^{2}}\right )^{\frac {1}{3}}}d x +\textit {\_C2}\right ]\] Mathematica raw input
DSolve[(x - y'[x]^2)*y''[x] == x^2 - y'[x],y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Inactive[Integrate][-((2^(1/3)*K[1])/(3*C[1] + K[1]^3 + Sqrt[9*C[1]^2 - 4*K[1]^3 + 6*C[1]*K[1]^3 + K[1]^6])^(1/3)) - (3*C[1] + K[1]^3 + Sqrt[9*C[1]^2 - 4*K[1]^3 + 6*C[1]*K[1]^3 + K[1]^6])^(1/3)/2^(1/3), {K[1], 1, x}]}, {y[x] -> C[2] + Inactive[Integrate][(2^(1/3)*(2 + (2*I)*Sqrt[3])*K[2] + 2^(2/3)*(1 - I*Sqrt[3])*(3*C[1] + K[2]^3 + Sqrt[9*C[1]^2 - 4*K[2]^3 + 6*C[1]*K[2]^3 + K[2]^6])^(2/3))/(4*(3*C[1] + K[2]^3 + Sqrt[9*C[1]^2 - 4*K[2]^3 + 6*C[1]*K[2]^3 + K[2]^6])^(1/3)), {K[2], 1, x}]}, {y[x] -> C[2] + Inactive[Integrate][(2^(1/3)*(2 - (2*I)*Sqrt[3])*K[3] + 2^(2/3)*(1 + I*Sqrt[3])*(3*C[1] + K[3]^3 + Sqrt[9*C[1]^2 - 4*K[3]^3 + 6*C[1]*K[3]^3 + K[3]^6])^(2/3))/(4*(3*C[1] + K[3]^3 + Sqrt[9*C[1]^2 - 4*K[3]^3 + 6*C[1]*K[3]^3 + K[3]^6])^(1/3)), {K[3], 1, x}]}}
Maple raw input
dsolve((x-diff(y(x),x)^2)*diff(diff(y(x),x),x) = x^2-diff(y(x),x), y(x))
Maple raw output
[y(x) = Int(1/2*((-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(2/3)+4*x)/(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(1/3),x)+_C2, y(x) = Int(-1/4*(I*3^(1/2)*(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(2/3)-4*I*3^(1/2)*x+(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(2/3)+4*x)/(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(1/3),x)+_C2, y(x) = Int(1/4*(I*3^(1/2)*(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(2/3)-4*I*3^(1/2)*x-(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(2/3)-4*x)/(-4*x^3+12*_C1+4*(x^6-6*_C1*x^3-4*x^3+9*_C1^2)^(1/2))^(1/3),x)+_C2]