Internal problem ID [2281]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 35, page 157
Problem number: 9.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }-{y^{\prime }}^{3}-y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 251
dsolve(diff(y(x),x$2)=diff(y(x),x)^3+diff(y(x),x),y(x), singsol=all)
\begin{align*} y \left (x \right ) = \frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1} -1\right ) {\mathrm e}^{2 x} c_{1}}}{2}-\frac {c_{1} \arctan \left (\frac {\sqrt {c_{1}^{2}}\, \left ({\mathrm e}^{2 x}-\frac {1}{2 c_{1}}\right )}{\sqrt {-\left ({\mathrm e}^{2 x} c_{1} -1\right ) {\mathrm e}^{2 x} c_{1}}}\right )}{4 \sqrt {c_{1}^{2}}}-\frac {\sqrt {c_{1} \left (-c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{2 x}\right )}}{2}-\frac {c_{1} \arctan \left (\frac {\sqrt {c_{1}^{2}}\, \left ({\mathrm e}^{2 x}-\frac {1}{2 c_{1}}\right )}{\sqrt {c_{1} \left (-c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{2 x}\right )}}\right )}{4 \sqrt {c_{1}^{2}}}+c_{2} y \left (x \right ) = -\frac {\sqrt {-\left ({\mathrm e}^{2 x} c_{1} -1\right ) {\mathrm e}^{2 x} c_{1}}}{2}+\frac {c_{1} \arctan \left (\frac {\sqrt {c_{1}^{2}}\, \left ({\mathrm e}^{2 x}-\frac {1}{2 c_{1}}\right )}{\sqrt {-\left ({\mathrm e}^{2 x} c_{1} -1\right ) {\mathrm e}^{2 x} c_{1}}}\right )}{4 \sqrt {c_{1}^{2}}}+\frac {\sqrt {c_{1} \left (-c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{2 x}\right )}}{2}+\frac {c_{1} \arctan \left (\frac {\sqrt {c_{1}^{2}}\, \left ({\mathrm e}^{2 x}-\frac {1}{2 c_{1}}\right )}{\sqrt {c_{1} \left (-c_{1} {\mathrm e}^{4 x}+{\mathrm e}^{2 x}\right )}}\right )}{4 \sqrt {c_{1}^{2}}}+c_{2} \end{align*}
✓ Solution by Mathematica
Time used: 60.102 (sec). Leaf size: 71
DSolve[y''[x]==y'[x]^3+y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_2-i \log \left (\sqrt {-1+e^{2 (x+c_1)}}-e^{x+c_1}\right ) y(x)\to i \log \left (\sqrt {-1+e^{2 (x+c_1)}}-e^{x+c_1}\right )+c_2 \end{align*}