2.5.2 Problem 3

Internal problem ID [18551]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 32. Problems at page 89
Problem number : 3
Date solved : Saturday, February 22, 2025 at 09:26:01 PM
CAS classification : [_quadrature]

Solve

\begin{align*} 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y&=0 \end{align*}

Solving for the derivative gives these ODE’s to solve

\begin{align*} \tag{1} y^{\prime }&=\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6} \\ \tag{2} y^{\prime }&=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \tag{3} y^{\prime }&=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ \end{align*}

Now each of the above is solved separately.

Solving Eq. (1)

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

\[ \int _{}^{y}\frac {6 \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}}{\left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{2}/{3}}-\left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}+1}d \tau = x +c_1 \]

We now need to find the singular solutions, these are found by finding for what values \((\frac {\left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}-\frac {1}{6})\) is zero. These give

\begin{align*} y&=\operatorname {RootOf}\left (-\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}+\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-1\right ) \\ \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

The solution \(\operatorname {RootOf}\left (-\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}+\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-1\right )\) will not be used

Solving Eq. (2)

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

\[ \int _{}^{y}\frac {12 \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}}{i \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{2}/{3}} \sqrt {3}-1-\left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{2}/{3}}-2 \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}-i \sqrt {3}}d \tau = x +c_2 \]

We now need to find the singular solutions, these are found by finding for what values \((-\frac {\left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {i \sqrt {3}\, \left (\frac {\left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}\right )}{2})\) is zero. These give

\begin{align*} y&=\operatorname {RootOf}\left (i \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}} \sqrt {3}-1-\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}-2 \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-i \sqrt {3}\right ) \\ \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

The solution \(\operatorname {RootOf}\left (i \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}} \sqrt {3}-1-\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}-2 \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-i \sqrt {3}\right )\) will not be used

Solving Eq. (3)

Unable to integrate (or intergal too complicated), and since no initial conditions are given, then the result can be written as

\[ \int _{}^{y}-\frac {12 \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}}{i \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{2}/{3}} \sqrt {3}+1+\left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{2}/{3}}+2 \left (-1+54 \tau +6 \sqrt {81 \tau ^{2}-3 \tau }\right )^{{1}/{3}}-i \sqrt {3}}d \tau = x +c_3 \]

We now need to find the singular solutions, these are found by finding for what values \((-\frac {\left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {i \sqrt {3}\, \left (\frac {\left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y +6 \sqrt {81 y^{2}-3 y}\right )^{{1}/{3}}}\right )}{2})\) is zero. These give

\begin{align*} y&=\operatorname {RootOf}\left (i \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}} \sqrt {3}+1+\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}+2 \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-i \sqrt {3}\right ) \\ \end{align*}

Now we go over each such singular solution and check if it verifies the ode itself and any initial conditions given. If it does not then the singular solution will not be used.

The solution \(\operatorname {RootOf}\left (i \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}} \sqrt {3}+1+\left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{2}/{3}}+2 \left (-1+54 \textit {\_Z} +6 \sqrt {3}\, \sqrt {27 \textit {\_Z}^{2}-\textit {\_Z}}\right )^{{1}/{3}}-i \sqrt {3}\right )\) will not be used

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & 2 {y^{\prime }}^{3}+{y^{\prime }}^{2}-y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \left [y^{\prime }=\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}, y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\textit {\_C1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =\int 1d x +\textit {\_C1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\textit {\_C1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\int \frac {y^{\prime }}{\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}+\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}}d x =x +\mathit {C1} , \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\mathit {C1} , \int \frac {y^{\prime }}{-\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{12}-\frac {1}{12 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}-\frac {1}{6}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}{6}-\frac {1}{6 \left (-1+54 y+6 \sqrt {-3 y+81 y^{2}}\right )^{{1}/{3}}}\right )}{2}}d x =x +\mathit {C1} \right \} \end {array} \]

Maple trace

`Methods for first order ODEs: 
   *** Sublevel 2 *** 
   Methods for first order ODEs: 
   -> Solving 1st order ODE of high degree, 1st attempt 
   trying 1st order WeierstrassP solution for high degree ODE 
   trying 1st order WeierstrassPPrime solution for high degree ODE 
   trying 1st order JacobiSN solution for high degree ODE 
   trying 1st order ODE linearizable_by_differentiation 
   trying differential order: 1; missing variables 
   <- differential order: 1; missing  x  successful`
 

Maple dsolve solution

Solving time : 0.019 (sec)
Leaf size : 385

dsolve(2*diff(y(x),x)^3+diff(y(x),x)^2-y(x) = 0,y(x),singsol=all)
 

\begin{align*} y \left (x \right ) &= 0 \\ -6 \sqrt {3}\, \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{3^{{2}/{3}}-\sqrt {3}\, \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}+3^{{1}/{3}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{2}/{3}}}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {-72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-c_{1} +x \right ) \sqrt {3}+3 i x -3 i c_{1}}{\sqrt {3}+3 i} &= 0 \\ \frac {72 \left (\int _{}^{y \left (x \right )}\frac {\left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}}{\left (-i 3^{{5}/{6}}+3^{{1}/{3}}-2 \,3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right ) \left (3^{{1}/{3}}+3^{{1}/{6}} \left (18 \sqrt {27 \textit {\_a}^{2}-\textit {\_a}}+\left (54 \textit {\_a} -1\right ) \sqrt {3}\right )^{{1}/{3}}\right )}d \textit {\_a} \right )+\left (-x +c_{1} \right ) \sqrt {3}+3 i x -3 i c_{1}}{-\sqrt {3}+3 i} &= 0 \\ \end{align*}

✗Mathematica DSolve solution

Solving time : 0.0 (sec)
Leaf size : 0

DSolve[{2*D[y[x],x]^3+D[y[x],x]^2-y[x]==0,{}},y[x],x,IncludeSingularSolutions->True]
 

Timed out

✓Sympy solution

Solving time : 63.755 (sec)
Leaf size : 343

Python version: 3.13.1 (main, Dec  4 2024, 18:05:56) [GCC 14.2.1 20240910] 
Sympy version 1.13.3
 

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + 2*Derivative(y(x), x)**3 + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

[Eq(-6*(sqrt(3) - I)*Integral((-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) 
+ 1)**(1/3)/(((-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) - 
1)*(sqrt(3)*(-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) + 
I*(-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) + 2*I)), (_y, 
y(x))), C1 - x), Eq(-6*(sqrt(3) + I)*Integral((-54*_y + 
6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3)/(((-54*_y + 
6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) - 1)*(sqrt(3)*(-54*_y + 
6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) - I*(-54*_y + 
6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) - 2*I)), (_y, y(x))), C1 - 
x), Eq(Integral((-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 
1)**(1/3)/((-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(2/3) + 
(-54*_y + 6*sqrt(3)*sqrt(_y*(27*_y - 1)) + 1)**(1/3) + 1), (_y, 
y(x))), C1 - x/6)]
 

\[ \left [ - 6 \left (\sqrt {3} - i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 2 i\right )}\, dy = C_{1} - x, \ - 6 \left (\sqrt {3} + i\right ) \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 1\right ) \left (\sqrt {3} \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - i \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} - 2 i\right )}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {\sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1}}{\left (- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1\right )^{\frac {2}{3}} + \sqrt [3]{- 54 y + 6 \sqrt {3} \sqrt {y \left (27 y - 1\right )} + 1} + 1}\, dy = C_{1} - \frac {x}{6}\right ] \]