29.3.9 problem 9

Internal problem ID [7248]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 3. Linear First-Order Equations. page 403
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 04:26:12 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

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

\[ y = \frac {x^{2}}{\sqrt {-x^{2}+1}}+\frac {c_1}{\sqrt {x -1}\, \sqrt {x +1}} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 4.730 (sec). Leaf size: 20

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

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