Internal problem ID [10127]
Internal file name [OUTPUT/9074_Monday_June_06_2022_06_22_15_AM_68249505/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1806 (book 6.215).
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_ode_missing_x"
Maple gives the following as the ode type
[[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]
\[ \boxed {\left (4 y^{3}-a y-b \right ) \left (y^{\prime \prime }+f y^{\prime }\right )-\left (6 y^{2}-\frac {a}{2}\right ) {y^{\prime }}^{2}=0} \]
This is missing independent variable second order ode. Solved by reduction of order by using substitution which makes the dependent variable \(y\) an independent variable. Using \begin {align*} y' &= p(y) \end {align*}
Then \begin {align*} y'' &= \frac {dp}{dx}\\ &= \frac {dy}{dx} \frac {dp}{dy}\\ &= p \frac {dp}{dy} \end {align*}
Hence the ode becomes \begin {align*} \left (4 y^{3}-y a -b \right ) p \left (y \right ) \left (\frac {d}{d y}p \left (y \right )\right )+\left (4 f \,y^{3}-y a f -b f -6 p \left (y \right ) y^{2}+\frac {a p \left (y \right )}{2}\right ) p \left (y \right ) = 0 \end {align*}
Which is now solved as first order ode for \(p(y)\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} \frac {d}{d y}p \left (y \right ) + p(y)p \left (y \right ) &= q(y) \end {align*}
Where here \begin {align*} p(y) &=-\frac {-12 y^{2}+a}{2 \left (-4 y^{3}+y a +b \right )}\\ q(y) &=\frac {8 f \,y^{3}-2 y a f -2 b f}{-8 y^{3}+2 y a +2 b} \end {align*}
Hence the ode is \begin {align*} \frac {d}{d y}p \left (y \right )-\frac {\left (-12 y^{2}+a \right ) p \left (y \right )}{2 \left (-4 y^{3}+y a +b \right )} = \frac {8 f \,y^{3}-2 y a f -2 b f}{-8 y^{3}+2 y a +2 b} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {-12 y^{2}+a}{2 \left (-4 y^{3}+y a +b \right )}d y} \\ &= \frac {1}{\sqrt {4 y^{3}-y a -b}} \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}}\left ( \mu p\right ) &= \left (\mu \right ) \left (\frac {8 f \,y^{3}-2 y a f -2 b f}{-8 y^{3}+2 y a +2 b}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}y}} \left (\frac {p}{\sqrt {4 y^{3}-y a -b}}\right ) &= \left (\frac {1}{\sqrt {4 y^{3}-y a -b}}\right ) \left (\frac {8 f \,y^{3}-2 y a f -2 b f}{-8 y^{3}+2 y a +2 b}\right )\\ \mathrm {d} \left (\frac {p}{\sqrt {4 y^{3}-y a -b}}\right ) &= \left (-\frac {f}{\sqrt {4 y^{3}-y a -b}}\right )\, \mathrm {d} y \end {align*}
Integrating gives \begin {align*} \frac {p}{\sqrt {4 y^{3}-y a -b}} &= \int {-\frac {f}{\sqrt {4 y^{3}-y a -b}}\,\mathrm {d} y}\\ \frac {p}{\sqrt {4 y^{3}-y a -b}} &= -\frac {2 i f \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {y -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}{-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{4}-\frac {3 a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{4}-\frac {3 a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3 \sqrt {4 y^{3}-y a -b}} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\frac {1}{\sqrt {4 y^{3}-y a -b}}\) results in \begin {align*} p \left (y \right ) &= -\frac {2 i f \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right ) \sqrt {-\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}\, \sqrt {\frac {y -\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}{-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{4}-\frac {3 a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}}}\, \sqrt {\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (y +\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{12}+\frac {a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ) \sqrt {3}}{\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}}}}{3}, \sqrt {-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{-\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{4}-\frac {3 a}{4 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}{6}-\frac {a}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}}\right )}{2}}}\right )}{3}+c_{1} \sqrt {4 y^{3}-y a -b} \end {align*}
which simplifies to \begin {align*} p \left (y \right ) &= \frac {3 \sqrt {2}\, \left (\left (\frac {2 i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}\, a}{3}+\left (-\frac {\left (1+i \sqrt {3}\right ) \sqrt {-3 a^{3}+81 b^{2}}}{3}-3 i \sqrt {3}\, b +4 y a -3 b \right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-i \sqrt {3}\, a^{2}+a^{2}-4 y \sqrt {-3 a^{3}+81 b^{2}}-36 y b \right ) f \sqrt {\frac {\left (-i-\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (i-\sqrt {3}\right ) a}{-\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}}\, \sqrt {\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-6 y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}+3 a}{i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 i \sqrt {3}\, a +3 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+9 a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {3}{4}} \sqrt {2}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}}{6 \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}}, \frac {\sqrt {2}\, 3^{\frac {1}{4}} \sqrt {-\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}}{\sqrt {\left (-\sqrt {3}+3 i\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+9 \left (i+\frac {\sqrt {3}}{3}\right ) a}}\right )+\frac {c_{1} \sqrt {4 y^{3}-y a -b}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}\, \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}}{3}\right )}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}} \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}} \end {align*}
For solution (1) found earlier, since \(p=y^{\prime }\) then we now have a new first order ode to solve which is \begin {align*} y^{\prime } = \frac {3 \sqrt {2}\, \left (\left (\frac {2 i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}\, a}{3}+\left (-\frac {\left (1+i \sqrt {3}\right ) \sqrt {-3 a^{3}+81 b^{2}}}{3}-3 i \sqrt {3}\, b +4 a y-3 b \right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-i \sqrt {3}\, a^{2}+a^{2}-4 y \sqrt {-3 a^{3}+81 b^{2}}-36 y b \right ) f \sqrt {\frac {\left (-i-\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (i-\sqrt {3}\right ) a}{-\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}}\, \sqrt {\frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-6 y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}+3 a}{i \sqrt {3}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 i \sqrt {3}\, a +3 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+9 a}}\, \operatorname {EllipticF}\left (\frac {3^{\frac {3}{4}} \sqrt {2}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}}{6 \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}}, \frac {\sqrt {2}\, 3^{\frac {1}{4}} \sqrt {-\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}}{\sqrt {\left (-\sqrt {3}+3 i\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+9 \left (i+\frac {\sqrt {3}}{3}\right ) a}}\right )+\frac {c_{1} \sqrt {4 y^{3}-a y-b}\, \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}\, \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}}{3}\right )}{2 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}} \sqrt {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}\, \sqrt {\left (-i+\sqrt {3}\right ) \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-12 i y \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}-3 \left (\sqrt {3}+i\right ) a}} \end {align*}
Integrating both sides gives \begin {align*} \text {Expression too large to display} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Warning, solution could not be verified
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville <- 2nd_order Liouville successful`
✓ Solution by Maple
Time used: 19.86 (sec). Leaf size: 257
dsolve((4*y(x)^3-a*y(x)-b)*(diff(diff(y(x),x),x)+f*diff(y(x),x))-(6*y(x)^2-1/2*a)*diff(y(x),x)^2=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= \frac {\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{6 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= \frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {3}\, a -\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}-3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ y \left (x \right ) &= -\frac {-i \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}} \sqrt {3}+3 i \sqrt {3}\, a +\left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {2}{3}}+3 a}{12 \left (27 b +3 \sqrt {-3 a^{3}+81 b^{2}}\right )^{\frac {1}{3}}} \\ c_{1} {\mathrm e}^{-f x}-c_{2} +\int _{}^{y \left (x \right )}\frac {1}{\sqrt {4 \textit {\_a}^{3}-a \textit {\_a} -b}}d \textit {\_a} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 10.455 (sec). Leaf size: 438
DSolve[(a/2 - 6*y[x]^2)*y'[x]^2 + (-b - a*y[x] + 4*y[x]^3)*(f[x]*y'[x] + y''[x]) == 0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {2 \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,1\right ]}} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]}} \left (y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-y(x)}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]}}\right ),\frac {\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,1\right ]-\text {Root}\left [4 \text {$\#$1}^3-a \text {$\#$1}-b\&,3\right ]}\right )}{\sqrt {a y(x)+b-4 y(x)^3} \sqrt {\frac {y(x)-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}{\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,2\right ]-\text {Root}\left [4 \text {$\#$1}^3-\text {$\#$1} a-b\&,3\right ]}}}=\int _1^x-\sqrt {2} \exp \left (-\int _1^{K[1]}f(K[1])dK[1]\right ) c_1dK[1]+c_2,y(x)\right ] \]