Internal problem ID [4267]
Internal file name [OUTPUT/3760_Sunday_June_05_2022_10_47_52_AM_9668248/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1046.
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]
\[ \boxed {{y^{\prime }}^{3}+\left (-3 x +1\right ) {y^{\prime }}^{2}-x \left (-3 x +1\right ) y^{\prime }=x^{3}+1} \] 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 (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3} \tag {1} \\ y^{\prime }&=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {x -\frac {1}{3}}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {2 x -\frac {2}{3}}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {x -\frac {1}{3}}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {2 x -\frac {2}{3}}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\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*} y = \int \frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{1} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+6 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-2 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{6 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} y = \int \frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}-\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+12 x -4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{2} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int \frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}-\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+12 x -4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{2} \\ \end{align*}
Verification of solutions
\[ y = \int \frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}-\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}+12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+12 x -4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} y = \int -\frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}+\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}-12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{3} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \int -\frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}+\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}-12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{3} \\ \end{align*}
Verification of solutions
\[ y = \int -\frac {i \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}} \sqrt {3}+12 i \sqrt {3}\, x -4 i \sqrt {3}+\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {2}{3}}-12 x \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}+4 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}-12 x +4}{12 \left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}d x +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+\left (-3 x +1\right ) {y^{\prime }}^{2}-x \left (-3 x +1\right ) y^{\prime }=x^{3}+1 \\ \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 (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}, y^{\prime }=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\int \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}\right )d x +c_{1} \\ {} & \circ & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\int \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}\right )d x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int y^{\prime }d x =\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d x +c_{1} \\ {} & \circ & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & y=\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d x +c_{1} \\ \bullet & {} & \textrm {Set of solutions}\hspace {3pt} \\ {} & {} & \left \{y=\int \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}\right )d x +c_{1} , y=\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d x +c_{1} , y=\int \left (-\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}+x -\frac {1}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {x}{3}-\frac {1}{9}\right )}{\left (36 x +100+12 \sqrt {12 x^{3}-3 x^{2}+54 x +69}\right )^{\frac {1}{3}}}\right )}{2}\right )d 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 y(x) successful`
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 380
dsolve(diff(y(x),x)^3+(1-3*x)*diff(y(x),x)^2-x*(1-3*x)*diff(y(x),x)-1-x^3 = 0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {\left (\int \frac {\left (1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}+12 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}-\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}-1\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {\left (-1+i \sqrt {3}\right ) \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}+12 \left (x -\frac {1}{3}\right ) \left (i \sqrt {3}+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}+1\right )}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{12}+c_{1} \\ y \left (x \right ) &= \frac {\left (\int \frac {4+6 \left (-2+\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}\right ) x +\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {2}{3}}-2 \left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}{\left (12 \sqrt {3}\, \sqrt {4 x^{3}-x^{2}+18 x +23}+36 x +100\right )^{\frac {1}{3}}}d x \right )}{6}+c_{1} \\ \end{align*}
✓ Solution by Mathematica
Time used: 110.523 (sec). Leaf size: 379
DSolve[(y'[x])^3+(1-3*x)(y'[x])^2-x*(1-3*x)*y'[x]-1 -x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \int _1^x\frac {1}{6} \left (6 K[1]-2^{2/3} \sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}+\frac {2 \sqrt [3]{2} (3 K[1]-1)}{\sqrt [3]{-9 K[1]+3 \sqrt {12 K[1]^3-3 K[1]^2+54 K[1]+69}-25}}-2\right )dK[1]+c_1 \\ y(x)\to \int _1^x\frac {1}{12} \left (12 K[2]+2^{2/3} \left (1-i \sqrt {3}\right ) \sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}-\frac {2 i \sqrt [3]{2} \left (-i+\sqrt {3}\right ) (3 K[2]-1)}{\sqrt [3]{-9 K[2]+3 \sqrt {12 K[2]^3-3 K[2]^2+54 K[2]+69}-25}}-4\right )dK[2]+c_1 \\ y(x)\to \int _1^x\frac {1}{12} \left (12 K[3]+2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}+\frac {2 i \sqrt [3]{2} \left (i+\sqrt {3}\right ) (3 K[3]-1)}{\sqrt [3]{-9 K[3]+3 \sqrt {12 K[3]^3-3 K[3]^2+54 K[3]+69}-25}}-4\right )dK[3]+c_1 \\ \end{align*}