23.1.354 problem 340

Internal problem ID [4961]
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 : 340
Date solved : Tuesday, September 30, 2025 at 09:05:53 AM
CAS classification : [_linear]

\begin{align*} 4 \left (x^{2}+1\right ) y^{\prime }-4 x y-x^{2}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 23

ode:=4*(x^2+1)*diff(y(x),x)-4*x*y(x)-x^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (4 c_1 +\operatorname {arcsinh}\left (x \right )\right ) \sqrt {x^{2}+1}}{4}-\frac {x}{4} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 48

ode=4*(1+x^2)*D[y[x],x]-4*x*y[x]-x^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{4} \left (\sqrt {x^2+1} \text {arctanh}\left (\frac {x}{\sqrt {x^2+1}}\right )+4 c_1 \sqrt {x^2+1}-x\right ) \end{align*}

✓ Sympy. Time used: 3.403 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 - 4*x*y(x) + (4*x**2 + 4)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x^{2} \operatorname {asinh}{\left (x \right )}}{4 \left (x^{2} + 1\right )} + \frac {x}{4 \sqrt {x^{2} + 1}} - \int \frac {x y{\left (x \right )}}{\left (x^{2} + 1\right )^{\frac {3}{2}}}\, dx - \frac {\operatorname {asinh}{\left (x \right )}}{4 \left (x^{2} + 1\right )} = C_{1} \]