15.18.9 problem 9

Internal problem ID [3252]
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
Date solved : Monday, January 27, 2025 at 07:28:47 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&={y^{\prime }}^{3}+y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 73

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 {\arctan \left (\frac {2 c_{1} {\mathrm e}^{2 x}-1}{2 \sqrt {-\left (c_{1} {\mathrm e}^{2 x}-1\right ) {\mathrm e}^{2 x} c_{1}}}\right )}{2}+c_2 \\ y \left (x \right ) &= \frac {\arctan \left (\frac {2 c_{1} {\mathrm e}^{2 x}-1}{2 \sqrt {-\left (c_{1} {\mathrm e}^{2 x}-1\right ) {\mathrm e}^{2 x} c_{1}}}\right )}{2}+c_2 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.135 (sec). Leaf size: 67

DSolve[D[y[x],{x,2}]==D[y[x],x]^3+D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to c_2-i \text {arctanh}\left (\frac {e^{x+c_1}}{\sqrt {-1+e^{2 (x+c_1)}}}\right ) \\ y(x)\to i \text {arctanh}\left (\frac {e^{x+c_1}}{\sqrt {-1+e^{2 (x+c_1)}}}\right )+c_2 \\ \end{align*}