Internal problem ID [4263]
Internal file name [OUTPUT/3756_Sunday_June_05_2022_10_46_34_AM_38475086/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1041.
ODE order: 1.
ODE degree: 3.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {{y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2}=0} \] Solving the given ode for \(y^{\prime }\) results in \(3\) differential equations to solve. Each one of these will generate a solution. The equations generated are \begin {align*} y^{\prime }&=\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3} \tag {1} \\ y^{\prime }&=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \tag {3} \end {align*}
Now each one of the above ODE is solved.
Solving equation (1)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {6 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+4}d \textit {\_a} = x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {6 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+4}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+2 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+4}d \textit {\_a} = x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}-4-\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}-4-\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}-4-\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4+\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \int _{}^{y}-\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4+\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {12 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{i \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}} \sqrt {3}+4+\left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {2}{3}}-4 \left (8-108 \textit {\_a}^{2}+12 \sqrt {81 \textit {\_a}^{4}-12 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-4 i \sqrt {3}}d \textit {\_a} = x +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}-{y^{\prime }}^{2}+y^{2}=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 (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}, y^{\prime }=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{\int \frac {y^{\prime }}{\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}+\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{12}-\frac {1}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}+\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}{6}-\frac {2}{3 \left (8-108 y^{2}+12 \sqrt {-12 y^{2}+81 y^{4}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \right \} \end {array} \]
Maple trace
`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`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 465
dsolve(diff(y(x),x)^3-diff(y(x),x)^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ -3 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3^{\frac {2}{3}} 2^{\frac {1}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {2}{3}}+6}d \textit {\_a} \right )+x -c_{1} &= 0 \\ \frac {36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right ) \left (3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}-6\right )}d \textit {\_a} \right )+\left (x -c_{1} \right ) \left (1+i \sqrt {3}\right )}{1+i \sqrt {3}} &= 0 \\ \frac {i \left (x -c_{1} \right ) \sqrt {3}+36 \,3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (\int _{}^{y \left (x \right )}\frac {\left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}}{\left (-3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+6\right ) \left (-3 i \sqrt {3}+3^{\frac {5}{6}} 2^{\frac {2}{3}} \left (-27 \sqrt {3}\, \textit {\_a}^{2}+2 \sqrt {3}+9 \sqrt {27 \textit {\_a}^{4}-4 \textit {\_a}^{2}}\right )^{\frac {1}{3}}+3\right )}d \textit {\_a} \right )-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 47.889 (sec). Leaf size: 583
DSolve[(y'[x])^3 - (y'[x])^2 +y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}}{2^{2/3} \left (-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2\right )^{2/3}+2 \sqrt [3]{-27 K[1]^2+3 \sqrt {3} \sqrt {K[1]^2 \left (27 K[1]^2-4\right )}+2}+2 \sqrt [3]{2}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}}{-i 2^{2/3} \sqrt {3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[2]^2+3 \sqrt {3} \sqrt {K[2]^2 \left (27 K[2]^2-4\right )}+2}+2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}}{i 2^{2/3} \sqrt {3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}-2^{2/3} \left (-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2\right )^{2/3}+4 \sqrt [3]{-27 K[3]^2+3 \sqrt {3} \sqrt {K[3]^2 \left (27 K[3]^2-4\right )}+2}-2 i \sqrt [3]{2} \sqrt {3}-2 \sqrt [3]{2}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}