15.18.38 problem 38

Internal problem ID [3281]
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 : 38
Date solved : Monday, January 27, 2025 at 07:30:45 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 2 y y^{\prime \prime }&=y^{3}+2 {y^{\prime }}^{2} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.814 (sec). Leaf size: 15

dsolve([2*y(x)*diff(y(x),x$2)=y(x)^3+2*diff(y(x),x)^2,y(0) = -1, D(y)(0) = 0],y(x), singsol=all)
 

\[ y \left (x \right ) = \operatorname {RootOf}\left (2 \,\operatorname {arctanh}\left (\sqrt {\textit {\_Z} +1}\right )+x \right ) \]

✓ Solution by Mathematica

Time used: 60.210 (sec). Leaf size: 15

DSolve[{2*y[x]*D[y[x],{x,2}]==y[x]^3+2*D[y[x],x]^2,{y[0]==-1,Derivative[1][y][0] ==0}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to -\text {sech}^2\left (\frac {x}{2}\right ) \]