60.3.349 problem 1355

Internal problem ID [11359]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1355
Date solved : Monday, January 27, 2025 at 11:17:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {\left (x^{3}-1\right ) y^{\prime }}{x \left (x^{3}+1\right )}+\frac {x y}{x^{3}+1} \end{align*}

✓ Solution by Maple

Time used: 0.214 (sec). Leaf size: 54

dsolve(diff(diff(y(x),x),x) = -(x^3-1)/x/(x^3+1)*diff(y(x),x)+x/(x^3+1)*y(x),y(x), singsol=all)
 

\[ y = \frac {2 c_{1} x^{2} \operatorname {LegendreP}\left (-\frac {1}{3}, -\frac {2}{3}, \frac {-x^{3}+1}{x^{3}+1}\right ) \Gamma \left (\frac {2}{3}\right )}{3 \left (x^{3}+1\right )^{{1}/{3}} \left (-x^{3}\right )^{{1}/{3}}}+c_{2} \left (x^{3}+1\right )^{{1}/{3}} \]

✓ Solution by Mathematica

Time used: 0.470 (sec). Leaf size: 110

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

\[ y(x)\to \exp \left (\int _1^x\frac {6 K[1]^3-2}{4 \left (K[1]^4+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {K[2]^3-1}{K[2]^4+K[2]}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {6 K[1]^3-2}{4 \left (K[1]^4+K[1]\right )}dK[1]\right )dK[3]+c_1\right ) \]