81.3.14 problem 4-19

Internal problem ID [21513]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 4. Homogeneous differential equations.
Problem number : 4-19
Date solved : Thursday, October 02, 2025 at 07:46:24 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y+\sqrt {x^{2}+y^{2}}-x y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.199 (sec). Leaf size: 21

ode:=y(x)+(x^2+y(x)^2)^(1/2)-x*diff(y(x),x) = 0; 
ic:=[y(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.174 (sec). Leaf size: 14

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

\begin{align*} y(x)&\to \frac {1}{2} \left (x^2-1\right ) \end{align*}

✓ Sympy. Time used: 0.725 (sec). Leaf size: 8

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 = {y(1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

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