23.3.344 problem 347

Internal problem ID [6058]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 347
Date solved : Tuesday, September 30, 2025 at 02:20:43 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 59

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

\[ y = -\frac {-2 \sqrt {x +1}\, \left (x +c_2 \right ) \sqrt {x -1}+c_1 \left (-x^{3}+\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {x^{2}-1}+x \right )}{2 \sqrt {x +1}\, \sqrt {x -1}} \]

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 43

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

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

✓ Sympy. Time used: 0.309 (sec). Leaf size: 29

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

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