ODE
\[ 2 y(x) y''(x)=y'(x)^2+3 y(x)^4 \] ODE Classification
[[_2nd_order, _missing_x]]
Book solution method
TO DO
Mathematica ✓
cpu = 8.60131 (sec), leaf count = 285
\[\left \{\left \{y(x)\to \text {InverseFunction}\left [-\frac {2 i \text {$\#$1}^{3/2} \sqrt {(-1)^{5/6} \left (\frac {\sqrt [3]{-c_1}}{\text {$\#$1}}-1\right )} \sqrt {\frac {\left (-c_1\right ){}^{2/3}}{\text {$\#$1}^2}+\frac {\sqrt [3]{-c_1}}{\text {$\#$1}}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {i \sqrt [3]{-c_1}}{\text {$\#$1}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{-c_1} \sqrt {\text {$\#$1}^3+c_1}}\& \right ]\left [c_2+x\right ]\right \},\left \{y(x)\to \text {InverseFunction}\left [\frac {2 i \text {$\#$1}^{3/2} \sqrt {(-1)^{5/6} \left (\frac {\sqrt [3]{-c_1}}{\text {$\#$1}}-1\right )} \sqrt {\frac {\left (-c_1\right ){}^{2/3}}{\text {$\#$1}^2}+\frac {\sqrt [3]{-c_1}}{\text {$\#$1}}+1} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {i \sqrt [3]{-c_1}}{\text {$\#$1}}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{-c_1} \sqrt {\text {$\#$1}^3+c_1}}\& \right ]\left [c_2+x\right ]\right \}\right \}\]
Maple ✓
cpu = 0.075 (sec), leaf count = 49
\[ \left \{ \int ^{y \left ( x \right ) }\!{\frac {1}{\sqrt {{{\it \_a}}^{4}+{\it \_a}\,{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0,\int ^{y \left ( x \right ) }\!-{\frac {1}{\sqrt {{{\it \_a}}^{4}+{\it \_a}\,{\it \_C1}}}}{d{\it \_a}}-x-{\it \_C2}=0 \right \} \] Mathematica raw input
DSolve[2*y[x]*y''[x] == 3*y[x]^4 + y'[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> InverseFunction[((-2*I)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-C[1])^(1/3))/#1]/3^(1/4)], (-1)^(1/3)]*Sqrt[(-1)^(5/6)*(-1 + (-C[1])^(1/3)/#1)]*Sqrt[1 + (-C[1])^(2/3)/#1^2 + (-C[1])^(1/3)/#1]*#1^(3/2))/(3^(1/4)*(-C[1])^(1/3)*Sqrt[C[1] + #1^3]) & ][x + C[2]]}, {y[x] -> InverseFunction[((2*I)*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-C[1])^(1/3))/#1]/3^(1/4)], (-1)^(1/3)]*Sqrt[(-1)^(5/6)*(-1 + (-C[1])^(1/3)/#1)]*Sqrt[1 + (-C[1])^(2/3)/#1^2 + (-C[1])^(1/3)/#1]*#1^(3/2))/(3^(1/4)*(-C[1])^(1/3)*Sqrt[C[1] + #1^3]) & ][x + C[2]]}}
Maple raw input
dsolve(2*y(x)*diff(diff(y(x),x),x) = diff(y(x),x)^2+3*y(x)^4, y(x),'implicit')
Maple raw output
Intat(1/(_a^4+_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0, Intat(-1/(_a^4+_C1*_a)^(1/2),_a = y(x))-x-_C2 = 0