56.38.2 problem Ex 2

Internal problem ID [14294]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 62. Summary. Page 144
Problem number : Ex 2
Date solved : Thursday, October 02, 2025 at 09:30:15 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.158 (sec). Leaf size: 59

ode:=(-x^2+1)*diff(diff(y(x),x),x)-x*diff(y(x),x) = 2; 
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: 0.017 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to c_2-\frac {1}{4} \left (-2 \log \left (\sqrt {x^2-1}+x\right )+c_1\right ){}^2 \end{align*}

✓ Sympy. Time used: 8.783 (sec). Leaf size: 22

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

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