Internal problem ID [4266]
Internal file name [OUTPUT/3759_Sunday_June_05_2022_10_47_39_AM_48305456/index.tex]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 35
Problem number: 1044.
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]
\[ \boxed {{y^{\prime }}^{3}+\operatorname {a0} {y^{\prime }}^{2}+\operatorname {a1} y^{\prime }+\operatorname {a3} y=-\operatorname {a2}} \] 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 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\operatorname {a1}}{3}-\frac {\operatorname {a0}^{2}}{9}\right )}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a0}}{3} \tag {1} \\ y^{\prime }&=-\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a0}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a0}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a0}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2} \tag {2} \\ y^{\prime }&=-\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {\operatorname {a1} -\frac {\operatorname {a0}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\operatorname {a0}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {2 \operatorname {a1} -\frac {2 \operatorname {a0}^{2}}{3}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} y-108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \operatorname {a1}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \operatorname {a0} \operatorname {a3} y-54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +81 \operatorname {a3}^{2} y^{2}+162 \operatorname {a3} y \operatorname {a2} +12 \operatorname {a3} y \operatorname {a0}^{3}+81 \operatorname {a2}^{2}+12 \operatorname {a2} \,\operatorname {a0}^{3}}\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 (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}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 (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} &= x +c_{1} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {6 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} = x +c_{1} \] Verified OK.
Solving equation (2)
Integrating both sides gives \begin {align*} \int _{}^{y}\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a0}^{2}+12 \operatorname {a1}}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 (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a0}^{2}+12 \operatorname {a1}}d \textit {\_a} &= x +c_{2} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} -\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-4 \operatorname {a0}^{2}+12 \operatorname {a1}}d \textit {\_a} = x +c_{2} \] Verified OK.
Solving equation (3)
Integrating both sides gives \begin {align*} \int _{}^{y}-\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} +\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}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 (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} +\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} &= x +c_{3} \\ \end{align*}
Verification of solutions
\[ \int _{}^{y}-\frac {12 \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \sqrt {3}\, \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-4 i \sqrt {3}\, \operatorname {a0}^{2}+12 i \sqrt {3}\, \operatorname {a1} +\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+4 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \textit {\_a} \,\operatorname {a0}^{3} \operatorname {a3} +81 \textit {\_a}^{2} \operatorname {a3}^{2}-54 \textit {\_a} \operatorname {a0} \operatorname {a1} \operatorname {a3} +12 \operatorname {a2} \,\operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} -54 \operatorname {a1} \operatorname {a0} \operatorname {a2} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} = x +c_{3} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & {y^{\prime }}^{3}+\mathit {a0} {y^{\prime }}^{2}+\mathit {a1} y^{\prime }+\mathit {a3} y=-\mathit {a2} \\ \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 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}, y^{\prime }=-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}, y^{\prime }=-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}\right ] \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}}d x =\int 1d x +c_{1} \\ {} & \circ & \textrm {Cannot compute integral}\hspace {3pt} \\ {} & {} & \int \frac {y^{\prime }}{\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\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 (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} \\ \square & {} & \textrm {Solve the equation}\hspace {3pt} y^{\prime }=-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2} \\ {} & \circ & \textrm {Separate variables}\hspace {3pt} \\ {} & {} & \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}}=1 \\ {} & \circ & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\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 (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\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 (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}-\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}-\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}\right )}{2}}d x =x +c_{1} , \int \frac {y^{\prime }}{-\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{12}+\frac {3 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}-\frac {\mathit {a0}}{3}+\frac {\mathrm {I} \sqrt {3}\, \left (\frac {\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\right )^{\frac {1}{3}}}{6}+\frac {6 \left (\frac {\mathit {a1}}{3}-\frac {\mathit {a0}^{2}}{9}\right )}{\left (36 \mathit {a1} \mathit {a0} -108 \mathit {a3} y-108 \mathit {a2} -8 \mathit {a0}^{3}+12 \sqrt {12 \mathit {a1}^{3}-3 \mathit {a1}^{2} \mathit {a0}^{2}-54 \mathit {a1} \mathit {a0} \mathit {a3} y-54 \mathit {a1} \mathit {a0} \mathit {a2} +81 \mathit {a3}^{2} y^{2}+162 \mathit {a3} y \mathit {a2} +12 \mathit {a3} y \mathit {a0}^{3}+81 \mathit {a2}^{2}+12 \mathit {a2} \,\mathit {a0}^{3}}\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: 0.047 (sec). Leaf size: 981
dsolve(diff(y(x),x)^3+a0*diff(y(x),x)^2+a1*diff(y(x),x)+a2+a3*y(x) = 0,y(x), singsol=all)
\begin{align*} x -6 \left (\int _{}^{y \left (x \right )}\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-2 \operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+4 \operatorname {a0}^{2}-12 \operatorname {a1}}d \textit {\_a} \right )-c_{1} &= 0 \\ \frac {-12 \left (\int _{}^{y \left (x \right )}\frac {\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \left (\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1} \right ) \sqrt {3}-\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}-\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1}}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 (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}}{i \left (\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}+2 \operatorname {a0}^{2}-6 \operatorname {a1} \right ) \sqrt {3}+\left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {2}{3}}+\operatorname {a0} \left (36 \operatorname {a1} \operatorname {a0} -108 \operatorname {a3} \textit {\_a} -108 \operatorname {a2} -8 \operatorname {a0}^{3}+12 \sqrt {12 \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0}^{3}-3 \operatorname {a1}^{2} \operatorname {a0}^{2}-54 \operatorname {a1} \left (\operatorname {a3} \textit {\_a} +\operatorname {a2} \right ) \operatorname {a0} +81 \textit {\_a}^{2} \operatorname {a3}^{2}+162 \textit {\_a} \operatorname {a2} \operatorname {a3} +12 \operatorname {a1}^{3}+81 \operatorname {a2}^{2}}\right )^{\frac {1}{3}}-2 \operatorname {a0}^{2}+6 \operatorname {a1}}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: 0.0 (sec). Leaf size: 0
DSolve[(y'[x])^3 + a0*(y'[x])^2 +a1*y'[x]+a2 +a3*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
Timed out