60.3.347 problem 1364

Internal problem ID [11343]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1364
Date solved : Sunday, March 30, 2025 at 08:17:35 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x) = 1/x*(2*b*c*x^c*(x^2-1)+2*(a-1)*x^2-2*a)/(x^2-1)*diff(y(x),x)-(b^2*c^2*x^(2*c)*(x^2-1)+b*c*x^(c+2)*(2*a-c-1)-b*c*x^c*(2*a-c+1)+x^2*(a*(a-1)-v*(v+1))-a*(a+1))/x^2/(x^2-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{b \,x^{c}} x^{a} \left (\operatorname {LegendreQ}\left (v , x\right ) c_2 +\operatorname {LegendreP}\left (v , x\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.232 (sec). Leaf size: 35

ode=D[y[x],{x,2}] == -(((-(a*(1 + a)) + ((-1 + a)*a - v*(1 + v))*x^2 - b*(1 + 2*a - c)*c*x^c + b*(-1 + 2*a - c)*c*x^(2 + c) + b^2*c^2*x^(2*c)*(-1 + x^2))*y[x])/(x^2*(-1 + x^2))) + ((-2*a + 2*(-1 + a)*x^2 + 2*b*c*x^c*(-1 + x^2))*D[y[x],x])/(x*(-1 + x^2)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (x^c\right )^{a/c} e^{b x^c} (c_1 \operatorname {LegendreP}(v,x)+c_2 \operatorname {LegendreQ}(v,x)) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
v = symbols("v") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) - (-2*a + 2*b*c*x**c*(x**2 - 1) + x**2*(2*a - 2))*Derivative(y(x), x)/(x*(x**2 - 1)) + (-a*(a + 1) + b**2*c**2*x**(2*c)*(x**2 - 1) - b*c*x**c*(2*a - c + 1) + b*c*x**(c + 2)*(2*a - c - 1) + x**2*(a*(a - 1) - v*(v + 1)))*y(x)/(x**2*(x**2 - 1)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None