29.13.2 problem 356

Internal problem ID [4956]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 13
Problem number : 356
Date solved : Tuesday, March 04, 2025 at 07:34:47 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 25

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

\[ y \left (x \right ) = \frac {-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right )+c_{1}}{\sqrt {x^{2}+1}} \]

✓ Mathematica. Time used: 0.064 (sec). Leaf size: 31

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

\[ y(x)\to \frac {-a \text {arctanh}\left (\sqrt {x^2+1}\right )+c_1}{\sqrt {x^2+1}} \]

✓ Sympy. Time used: 3.320 (sec). Leaf size: 42

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

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