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 = 0.664688 (sec), leaf count = 721
\[\left \{\left \{y(x)\to \frac {c_1^2 \left (3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}\right ){}^{2/3}-4 c_1 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}+16}{2 c_1^2 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}}\right \},\left \{y(x)\to \frac {i \left (c_1 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}+4\right ) \left (\left (\sqrt {3}+i\right ) c_1 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}-4 \sqrt {3}+4 i\right )}{4 c_1^2 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}}\right \},\left \{y(x)\to -\frac {i \left (c_1 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}+4\right ) \left (\left (\sqrt {3}-i\right ) c_1 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}-4 \left (\sqrt {3}+i\right )\right )}{4 c_1^2 \sqrt [3]{3 \sqrt {\frac {\left (c_2+x\right ){}^2 \left (9 c_1^4 x^2+18 c_2 c_1^4 x+9 c_2^2 c_1^4+128\right )}{c_1^2}}+9 c_1 \left (c_2+x\right ){}^2+\frac {64}{c_1^3}}}\right \}\right \}\]
Maple ✓
cpu = 0.084 (sec), leaf count = 57
\[ \left \{ -{\frac {2\,{\it \_C1}\,y \left ( x \right ) +8}{3\,{{\it \_C1}}^{2}}\sqrt {{\it \_C1}\,y \left ( x \right ) -2}}-x-{\it \_C2}=0,{\frac {2\,{\it \_C1}\,y \left ( x \right ) +8}{3\,{{\it \_C1}}^{2}}\sqrt {{\it \_C1}\,y \left ( x \right ) -2}}-x-{\it \_C2}=0 \right \} \] 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] -> ((-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 + 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))}}
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),'implicit')
Maple raw output
-2/3*(_C1*y(x)-2)^(1/2)*(_C1*y(x)+4)/_C1^2-x-_C2 = 0, 2/3*(_C1*y(x)-2)^(1/2)*(_C1*y(x)+4)/_C1^2-x-_C2 = 0