60.3.218 problem 1234

Internal problem ID [11214]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1234
Date solved : Wednesday, March 05, 2025 at 01:45:47 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }+x y^{\prime }+2&=0 \end{align*}

✓ Maple. Time used: 0.340 (sec). Leaf size: 59

ode:=(x^2-1)*diff(diff(y(x),x),x)+x*diff(y(x),x)+2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\int -\frac {-2 \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {x -1}\, \sqrt {x +1}+c_{1} \left (x^{2}-1\right )}{\left (x -1\right )^{{3}/{2}} \left (x +1\right )^{{3}/{2}}}d x +c_{2} \]

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 31

ode=2 + x*D[y[x],x] + (-1 + x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2-\frac {1}{4} \left (-2 \log \left (\sqrt {x^2-1}+x\right )+c_1\right ){}^2 \]

✓ Sympy. Time used: 13.174 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = C_{1} - \frac {\left (C_{2} + 2 \log {\left (x + \sqrt {x^{2} - 1} \right )}\right )^{2}}{4} \]