60.3.355 problem 1361

Internal problem ID [11365]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1361
Date solved : Monday, January 27, 2025 at 11:18:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (a \left (a +1\right )-a \,x^{2} \left (a +3\right )\right ) y}{x^{2} \left (x^{2}-1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 33

dsolve(diff(diff(y(x),x),x) = 2*x/(x^2-1)*diff(y(x),x)-(a*(a+1)-a*x^2*(a+3))/x^2/(x^2-1)*y(x),y(x), singsol=all)
 

\[ y = c_{1} x^{-a}+c_{2} x^{a +1} \left (2 a \,x^{2}+x^{2}-2 a -3\right ) \]

✓ Solution by Mathematica

Time used: 0.852 (sec). Leaf size: 67

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

\[ y(x)\to c_2 \exp \left (\int \frac {2 a^2 \left (x^2-1\right )+a \left (7 x^2-5\right )+3 \left (x^2-1\right )}{x \left (2 a \left (x^2-1\right )+x^2-3\right )} \, dx\right )+c_1 x^{-a} \]