23.1.200 problem 197

Internal problem ID [4807]
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 : 197
Date solved : Tuesday, September 30, 2025 at 08:41:26 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 26

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

\[ \frac {-c_1 \,x^{2}+\sqrt {x^{2}+y^{2}}+y}{x^{2}} = 0 \]

✓ Mathematica. Time used: 0.185 (sec). Leaf size: 13

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

\begin{align*} y(x)&\to x \sinh (\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.724 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = - x \sinh {\left (C_{1} - \log {\left (x \right )} \right )} \]