60.3.352 problem 1358

Internal problem ID [11362]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1358
Date solved : Monday, January 27, 2025 at 11:18:01 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 20

dsolve(diff(diff(y(x),x),x) = 1/x*(x^2-2)/(x^2-1)*diff(y(x),x)-(x^2-2)/x^2/(x^2-1)*y(x),y(x), singsol=all)
 

\[ y = x \left (c_{1} +c_{2} \ln \left (x +\sqrt {x^{2}-1}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.395 (sec). Leaf size: 67

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

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