Internal problem ID [4249]
Book: Ordinary differential equations and their solutions. By George Moseley Murphy.
1960
Section: Various 34
Problem number: 1024.
ODE order: 1.
ODE degree: 3.
CAS Maple gives this as type [_quadrature]
\[ \boxed {{y^{\prime }}^{3}+y^{\prime }-{\mathrm e}^{y}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 261
dsolve(diff(y(x),x)^3+diff(y(x),x) = exp(y(x)),y(x), singsol=all)
\begin{align*} x -\left (\int _{}^{y \left (x \right )}\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} \right )-c_{1} = 0 x -\left (\int _{}^{y \left (x \right )}\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (1+i \sqrt {3}\right ) \left (\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}+\sqrt {3}-3 i\right ) \left (-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}-3 i+\sqrt {3}\right )}d \textit {\_a} \right )-c_{1} = 0 x -\left (\int _{}^{y \left (x \right )}-\frac {12 \left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}}{\left (-1+i \sqrt {3}\right ) \left (\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}+\sqrt {3}+3 i\right ) \left (-\left (108 \,{\mathrm e}^{\textit {\_a}}+12 \sqrt {12+81 \,{\mathrm e}^{2 \textit {\_a}}}\right )^{\frac {1}{3}}+3 i+\sqrt {3}\right )}d \textit {\_a} \right )-c_{1} = 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*}