29.10.18 problem 284

Internal problem ID [4884]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 10
Problem number : 284
Date solved : Sunday, March 30, 2025 at 04:08:54 AM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 66

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

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

✓ Mathematica. Time used: 0.119 (sec). Leaf size: 34

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

\[ y(x)\to -\sqrt {1-x^2} \arcsin (x)+c_1 \sqrt {x^2-1}+x \]

✓ Sympy. Time used: 0.463 (sec). Leaf size: 99

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

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