23.1.373 problem 358

Internal problem ID [4980]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 1. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF FIRST DEGREE, page 223
Problem number : 358
Date solved : Tuesday, September 30, 2025 at 09:08:42 AM
CAS classification : [_linear]

\begin{align*} x \left (x^{2}+1\right ) y^{\prime }&=a \,x^{3}+y \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 17

ode:=x*(x^2+1)*diff(y(x),x) = a*x^3+y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 69

ode=x*(1+x^2)*D[y[x],x]==a*x^3+y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {1}{K[1]^3+K[1]}dK[1]\right ) \left (\int _1^x\frac {a \exp \left (-\int _1^{K[2]}\frac {1}{K[1]^3+K[1]}dK[1]\right ) K[2]^2}{K[2]^2+1}dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.667 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**3 + x*(x**2 + 1)*Derivative(y(x), x) - y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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