34.22 problem 1024

Internal problem ID [4249]
Internal file name [OUTPUT/3742_Sunday_June_05_2022_10_35_36_AM_7236292/index.tex]

Book: Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section: Various 34
Problem number: 1024.
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]

\[ \boxed {{y^{\prime }}^{3}+y^{\prime }-{\mathrm e}^{y}=0} \] 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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}} \tag {1} \\ y^{\prime }&=-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \,{\mathrm e}^{2 y}}\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 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}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 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} &= x +c_{1} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {6 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{1} \] Verified OK.

Solving equation (2)

Integrating both sides gives \begin {align*} \int _{}^{y}\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12}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 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12}d \textit {\_a} &= x +c_{2} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12}d \textit {\_a} = x +c_{2} \] Verified OK.

Solving equation (3)

Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}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 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} &= x +c_{3} \\ \end{align*}

Verification of solutions

\[ \int _{}^{y}-\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+12 i \sqrt {3}+\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} = x +c_{3} \] Verified OK.

34.22.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+y^{\prime }-{\mathrm e}^{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 (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}, y^{\prime }=-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 {\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}-\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{12}+\frac {1}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\right )^{\frac {1}{3}}}{6}+\frac {2}{\left (108 \,{\mathrm e}^{y}+12 \sqrt {12+81 \left ({\mathrm e}^{y}\right )^{2}}\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: 0.015 (sec). Leaf size: 233

dsolve(diff(y(x),x)^3+diff(y(x),x) = exp(y(x)),y(x), singsol=all)
 

\begin{align*} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-12}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}-6-6 i \sqrt {3}}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 (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {2}{3}}+\left (\sqrt {3}+3 i\right )^{2}}d \textit {\_a} \right )+i \left (x -c_{1} \right ) \sqrt {3}+c_{1} -x}{-1+i \sqrt {3}} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 168.19 (sec). Leaf size: 1244

DSolve[(y'[x])^3 +y'[x]==Exp[ y[x]],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{36} \left (\frac {e^{-\text {$\#$1}} \left (2^{2/3} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}} \sqrt {81 e^{2 \text {$\#$1}}+12}-9\ 2^{2/3} e^{\text {$\#$1}} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}+4\ 3^{2/3}\right )}{\left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}}-12 \sqrt [3]{6} \arctan \left (\frac {6^{2/3} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}{\sqrt [3]{2} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}-2 \sqrt [3]{3}}\right )\right )+\frac {e^{-\text {$\#$1}}}{3\ 6^{2/3}}\&\right ]\left [-\frac {x}{6^{2/3}}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {e^{-\text {$\#$1}}}{6\ 2^{2/3} 3^{5/6}}-\frac {1}{144} i \left (\frac {e^{-\text {$\#$1}} \left (-12 i \sqrt [3]{2} \sqrt [6]{3} e^{\text {$\#$1}} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3} \arctan \left (\frac {6^{2/3} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}{\sqrt [3]{2} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}-2 \sqrt [3]{3}}\right )-3 i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}-3 i\right ) e^{\text {$\#$1}} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}+2^{2/3} 3^{5/6} \sqrt {27 e^{2 \text {$\#$1}}+4} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}+i 2^{2/3} \sqrt [3]{3} \sqrt {27 e^{2 \text {$\#$1}}+4} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}+4 i \sqrt {3}-12\right )}{\left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}}-24 \sqrt [3]{2} \sqrt [6]{3} \text {arctanh}\left (\frac {\sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}{\sqrt [3]{2} \sqrt [6]{3}}\right )-12 \sqrt [3]{2} \sqrt [6]{3} \text {arctanh}\left (\frac {\sqrt [3]{2} 3^{2/3} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}+6}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}\right )\right )\&\right ]\left [\frac {x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [-\frac {e^{-\text {$\#$1}}}{6\ 2^{2/3} 3^{5/6}}+\frac {1}{144} i \left (\frac {e^{-\text {$\#$1}} \left (12 i \sqrt [3]{2} \sqrt [6]{3} e^{\text {$\#$1}} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3} \arctan \left (\frac {6^{2/3} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}{\sqrt [3]{2} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}-2 \sqrt [3]{3}}\right )+3 i 2^{2/3} \sqrt [3]{3} \left (\sqrt {3}+3 i\right ) e^{\text {$\#$1}} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}+2^{2/3} 3^{5/6} \sqrt {27 e^{2 \text {$\#$1}}+4} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}-i 2^{2/3} \sqrt [3]{3} \sqrt {27 e^{2 \text {$\#$1}}+4} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}-4 i \sqrt {3}-12\right )}{\left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}}-24 \sqrt [3]{2} \sqrt [6]{3} \text {arctanh}\left (\frac {\sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}{\sqrt [3]{2} \sqrt [6]{3}}\right )-12 \sqrt [3]{2} \sqrt [6]{3} \text {arctanh}\left (\frac {\sqrt [3]{2} 3^{2/3} \left (\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}\right )^{2/3}+6}{2^{2/3} 3^{5/6} \sqrt [3]{\sqrt {81 e^{2 \text {$\#$1}}+12}-9 e^{\text {$\#$1}}}}\right )\right )\&\right ]\left [\frac {x}{2\ 2^{2/3} 3^{5/6}}+c_1\right ] \\ \end{align*}