Internal problem ID [4268]
Internal file name [OUTPUT/3761_Sunday_June_05_2022_10_47_58_AM_15168427/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1047.
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]
\[ \boxed {{y^{\prime }}^{3}-y {y^{\prime }}^{2}+y^{2}=0} \] 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 (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3} \tag {1} \\ y^{\prime }&=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\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 (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}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 (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+4 \textit {\_a}^{2}}d \textit {\_a} = x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{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 (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+4 \textit {\_a}^{2}-4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-4 \textit {\_a}^{2}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{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 (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-4 \textit {\_a}^{2}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {12 \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{4 i \textit {\_a}^{2} \sqrt {3}+4 \textit {\_a} \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-4 \textit {\_a}^{2}-i \sqrt {3}\, \left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}-\left (-108 \textit {\_a}^{2}+8 \textit {\_a}^{3}+12 \sqrt {-12 \textit {\_a}^{5}+81 \textit {\_a}^{4}}\right )^{\frac {2}{3}}}d \textit {\_a} = x +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}-y {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 (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}, y^{\prime }=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\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 (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\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 (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\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 (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}+\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{12}-\frac {y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}+\frac {y}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}{6}-\frac {2 y^{2}}{3 \left (-108 y^{2}+8 y^{3}+12 \sqrt {81 y^{4}-12 y^{5}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \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`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 420
dsolve(diff(y(x),x)^3-y(x)*diff(y(x),x)^2+y(x)^2 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= 0 \\ x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {2}{3}}+2 \left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}} \textit {\_a} +4 \textit {\_a}^{2}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (i \textit {\_a} \sqrt {3}+\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right ) \left (\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}-2 \textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+x -c_{1}}{1+i \sqrt {3}} &= 0 \\ \frac {12 \left (\int _{}^{y \left (x \right )}\frac {\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}}{\left (-i \textit {\_a} \sqrt {3}+\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+\textit {\_a} \right ) \left (-\left (8 \textit {\_a}^{3}-108 \textit {\_a}^{2}+12 \sqrt {3}\, \sqrt {-4 \textit {\_a}^{5}+27 \textit {\_a}^{4}}\right )^{\frac {1}{3}}+2 \textit {\_a} \right )}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}-x +c_{1}}{-1+i \sqrt {3}} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 56.7 (sec). Leaf size: 653
DSolve[(y'[x])^3 -y[x]*(y'[x])^2+y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}}}{2 \sqrt [3]{2} K[1]^2+2 \sqrt [3]{2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}} K[1]+2^{2/3} \left (2 K[1]^3-27 K[1]^2+3 \sqrt {3} \sqrt {-K[1]^4 (4 K[1]-27)}\right )^{2/3}}dK[1]\&\right ]\left [\frac {x}{6}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}}}{2 i \sqrt [3]{2} \sqrt {3} K[2]^2-2 \sqrt [3]{2} K[2]^2+4 \sqrt [3]{2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}} K[2]-i 2^{2/3} \sqrt {3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}-2^{2/3} \left (2 K[2]^3-27 K[2]^2+3 \sqrt {3} \sqrt {-K[2]^4 (4 K[2]-27)}\right )^{2/3}}dK[2]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {\sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}}}{-2 i \sqrt [3]{2} \sqrt {3} K[3]^2-2 \sqrt [3]{2} K[3]^2+4 \sqrt [3]{2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}} K[3]+i 2^{2/3} \sqrt {3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}-2^{2/3} \left (2 K[3]^3-27 K[3]^2+3 \sqrt {3} \sqrt {-K[3]^4 (4 K[3]-27)}\right )^{2/3}}dK[3]\&\right ]\left [\frac {x}{12}+c_1\right ] \\ y(x)\to 0 \\ \end{align*}