ODE
\[ 2 y(x)^3 y''(x)+y(x)^2 y'(x)^2=2 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.843253 (sec), leaf count = 676
\[\left \{\left \{y(x)\to \frac {c_1{}^2 \left (3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}\right ){}^{2/3}-4 c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}+16}{2 c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}\right \},\left \{y(x)\to \frac {i \left (4+c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}\right ) \left (\left (\sqrt {3}+i\right ) c_1 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}-4 \sqrt {3}+4 i\right )}{4 c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}\right \},\left \{y(x)\to -\frac {1}{4} i \left (\left (\sqrt {3}-i\right ) \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}-\frac {16 \left (\sqrt {3}+i\right )}{c_1{}^2 \sqrt [3]{3 \sqrt {\frac {(x+c_2){}^2 \left (9 c_1{}^4 x^2+18 c_2 c_1{}^4 x+128+9 c_2{}^2 c_1{}^4\right )}{c_1{}^2}}+9 c_1 (x+c_2){}^2+\frac {64}{c_1{}^3}}}-\frac {4 \left (\sqrt {3}+i\right )}{c_1}+\frac {4 \left (\sqrt {3}-i\right )}{c_1}\right )\right \}\right \}\]
Maple ✓
cpu = 0.492 (sec), leaf count = 1021
\[\left [y \left (x \right ) = \frac {\left (\frac {\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}-\frac {4}{\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )^{2}+2}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (\frac {\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}-\frac {4}{\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )^{2}+2}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (-\frac {\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{4}+\frac {2}{\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}+\frac {4}{\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (-\frac {\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{4}+\frac {2}{\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}+\frac {4}{\left (-6 \textit {\_C1}^{2} \textit {\_C2} -6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (-\frac {\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{4}+\frac {2}{\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}+\frac {4}{\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2}{\textit {\_C1}}, y \left (x \right ) = \frac {\left (-\frac {\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{4}+\frac {2}{\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}{2}+\frac {4}{\left (6 \textit {\_C1}^{2} \textit {\_C2} +6 \textit {\_C1}^{2} x +2 \sqrt {9 \textit {\_C1}^{4} \textit {\_C2}^{2}+18 \textit {\_C1}^{4} \textit {\_C2} x +9 \textit {\_C1}^{4} x^{2}+128}\right )^{\frac {1}{3}}}\right )}{2}\right )^{2}+2}{\textit {\_C1}}\right ]\] Mathematica raw input
DSolve[y[x]^2*y'[x]^2 + 2*y[x]^3*y''[x] == 2,y[x],x]
Mathematica raw output
{{y[x] -> (16 - 4*C[1]*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3) + C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(2/3))/(2*C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3))}, {y[x] -> ((I/4)*(4 + C[1]*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3))*(4*I - 4*Sqrt[3] + (I + Sqrt[3])*C[1]*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3)))/(C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3))}, {y[x] -> (-1/4*I)*((4*(-I + Sqrt[3]))/C[1] - (4*(I + Sqrt[3]))/C[1] - (16*(I + Sqrt[3]))/(C[1]^2*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3)) + (-I + Sqrt[3])*(64/C[1]^3 + 9*C[1]*(x + C[2])^2 + 3*Sqrt[((x + C[2])^2*(128 + 9*x^2*C[1]^4 + 18*x*C[1]^4*C[2] + 9*C[1]^4*C[2]^2))/C[1]^2])^(1/3))}}
Maple raw input
dsolve(2*y(x)^3*diff(diff(y(x),x),x)+y(x)^2*diff(y(x),x)^2 = 2, y(x))
Maple raw output
[y(x) = ((1/2*(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)-4/(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3))^2+2)/_C1, y(x) = ((1/2*(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)-4/(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3))^2+2)/_C1, y(x) = ((-1/4*(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+2/(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+4/(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)))^2+2)/_C1, y(x) = ((-1/4*(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+2/(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+4/(-6*_C1^2*_C2-6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)))^2+2)/_C1, y(x) = ((-1/4*(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+2/(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)-1/2*I*3^(1/2)*(1/2*(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+4/(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)))^2+2)/_C1, y(x) = ((-1/4*(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+2/(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+1/2*I*3^(1/2)*(1/2*(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)+4/(6*_C1^2*_C2+6*_C1^2*x+2*(9*_C1^4*_C2^2+18*_C1^4*_C2*x+9*_C1^4*x^2+128)^(1/2))^(1/3)))^2+2)/_C1]