23.1.371 problem 356

Internal problem ID [4978]
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 : 356
Date solved : Tuesday, September 30, 2025 at 09:08:39 AM
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: 19

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

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

✓ Mathematica. Time used: 0.104 (sec). Leaf size: 67

ode=x*(1+x^2)*D[y[x],x]==a*x^2+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]}{K[2]^2+1}dK[2]+c_1\right ) \end{align*}

✓ Sympy. Time used: 1.632 (sec). Leaf size: 19

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

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