60.7.99 problem 1690 (book 6.99)

Internal problem ID [11688]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1690 (book 6.99)
Date solved : Tuesday, January 28, 2025 at 06:07:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime }+\left (-y+x y^{\prime }\right )^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.108 (sec). Leaf size: 37

dsolve(x^4*diff(diff(y(x),x),x)+(x*diff(y(x),x)-y(x))^3=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.197 (sec). Leaf size: 95

DSolve[(-y[x] + x*D[y[x],x])^3 + x^4*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -i x \log \left (\frac {e^{c_2}-\sqrt {e^{2 c_2}-8 i c_1 x^2}}{4 c_1 x}\right ) \\ y(x)\to -i x \log \left (\frac {\sqrt {e^{2 c_2}-8 i c_1 x^2}+e^{c_2}}{4 c_1 x}\right ) \\ \end{align*}