88.10.7 problem 7

Internal problem ID [24042]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 52
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:55:08 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 61

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

\[ y = -\frac {-4 \sqrt {x +1}\, \left (x +\frac {c_2}{2}\right ) \sqrt {x -1}+\left (-x^{3}+\ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {x^{2}-1}+x \right ) c_1}{2 \sqrt {x -1}\, \sqrt {x +1}} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 45

ode=(1-x^2)*D[y[x],{x,2}]+x*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} x \left (4+c_1 \sqrt {x^2-1}\right )+c_2 \end{align*}

✓ Sympy. Time used: 0.314 (sec). Leaf size: 31

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

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