89.6.43 problem 44

Internal problem ID [24426]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Miscellaneous Exercises at page 45
Problem number : 44
Date solved : Thursday, October 02, 2025 at 10:27:47 PM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (2\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 22

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

\[ y = \frac {\left (x +2 \ln \left (x -1\right )+1\right ) \left (x -1\right )}{x +1} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 30

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

\begin{align*} y(x)&\to \frac {(x-1) (x+2 \log (1-x)-2 i \pi +1)}{x+1} \end{align*}

✓ Sympy. Time used: 0.209 (sec). Leaf size: 26

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

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