60.3.369 problem 1375

Internal problem ID [11379]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1375
Date solved : Tuesday, January 28, 2025 at 06:04:46 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {2 x \left (n +1-2 a \right ) y^{\prime }}{x^{2}-1}-\frac {\left (4 a \,x^{2} \left (a -n \right )-\left (x^{2}-1\right ) \left (2 a +\left (v -n \right ) \left (v +n +1\right )\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 29

dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*(n+1-2*a)*diff(y(x),x)-(4*a*x^2*(a-n)-(x^2-1)*(2*a+(v-n)*(v+n+1)))/(x^2-1)^2*y(x),y(x), singsol=all)
 

\[ y = \left (\operatorname {LegendreP}\left (v , n , x\right ) c_{1} +\operatorname {LegendreQ}\left (v , n , x\right ) c_{2} \right ) \left (x^{2}-1\right )^{a -\frac {n}{2}} \]

✓ Solution by Mathematica

Time used: 0.068 (sec). Leaf size: 34

DSolve[D[y[x],{x,2}] == -(((4*a*(a - n)*x^2 - (2*a + (-n + v)*(1 + n + v))*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*(1 - 2*a + n)*x*D[y[x],x])/(-1 + x^2),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \left (x^2-1\right )^{a-\frac {n}{2}} (c_1 P_v^n(x)+c_2 Q_v^n(x)) \]