23.1.405 problem 392

Internal problem ID [5012]
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 : 392
Date solved : Tuesday, September 30, 2025 at 11:22:55 AM
CAS classification : [_linear]

\begin{align*} y^{\prime } \sqrt {x^{2}+1}&=2 x -y \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 33

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

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

✓ Sympy. Time used: 0.233 (sec). Leaf size: 34

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

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