89.2.23 problem 25

Internal problem ID [24288]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 27
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:07:32 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 (\sqrt {3}\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.204 (sec). Leaf size: 21

ode:=y(x)+(x^2+y(x)^2)^(1/2)-x*diff(y(x),x) = 0; 
ic:=[y(3^(1/2)) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= -\frac {x^{2}}{6}+\frac {3}{2} \\ y &= \frac {x^{2}}{2}-\frac {1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.181 (sec). Leaf size: 23

ode=(y[x]+Sqrt[x^2+y[x]^2] )-(x )*D[y[x],x]==0; 
ic={y[Sqrt[3]]==1}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \sinh \left (\text {arcsinh}\left (\frac {1}{\sqrt {3}}\right )+\log (x)-\frac {\log (3)}{2}\right ) \end{align*}

✗ Sympy

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(sqrt(3)): 1} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants