83.29.5 problem 5

Internal problem ID [19378]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (C) at page 107
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 01:35:52 PM
CAS classification : [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 43

dsolve(diff(y(x),x$2)=y(x)^3-y(x),y(x), singsol=all)
 

\[ y \left (x \right ) = c_{2} \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}\, \operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, x +2 c_{1} \right ) \sqrt {2}\, \sqrt {\frac {1}{c_{2}^{2}+1}}}{2}, c_{2}\right ) \]

✓ Solution by Mathematica

Time used: 60.160 (sec). Leaf size: 171

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

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