60.7.7 problem 1597 (6.7)

Internal problem ID [11596]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1597 (6.7)
Date solved : Monday, January 27, 2025 at 11:23:56 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }-a y^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 25

dsolve(diff(diff(y(x),x),x)-a*y(x)^3=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 61.202 (sec). Leaf size: 131

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

\begin{align*} y(x)\to -\frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\ y(x)\to \frac {i \sqrt [4]{2} \text {sn}\left (\left .-\frac {(1-i) \sqrt {\sqrt {a} \sqrt {c_1} (x+c_2){}^2}}{2^{3/4}}\right |-1\right )}{\sqrt {\frac {i \sqrt {a}}{\sqrt {c_1}}}} \\ \end{align*}