89.2.20 problem 20

Internal problem ID [24285]
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 : 20
Date solved : Thursday, October 02, 2025 at 10:07:15 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 80

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

\[ \frac {2^{\frac {x}{\sqrt {-x^{2}}}} {\left (\frac {\sqrt {-x^{2}}\, \sqrt {y^{2}-x^{2}}-x^{2}}{y}\right )}^{\frac {x}{\sqrt {-x^{2}}}} x^{\frac {\sqrt {-x^{2}}}{x}}-c_1 y}{y} = 0 \]

✓ Mathematica. Time used: 1.142 (sec). Leaf size: 118

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

\[ \text {Solve}\left [\frac {\sqrt {\frac {y(x)^2}{x^2}-1} \left (\log \left (\sqrt {\frac {y(x)}{x}+1}-1\right )+\log \left (\sqrt {\frac {y(x)}{x}+1}+1\right )\right )}{\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}}-2 \arctan \left (\frac {y(x)}{x}-\sqrt {\frac {y(x)}{x}-1} \sqrt {\frac {y(x)}{x}+1}\right )=-\log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 1.142 (sec). Leaf size: 12

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

\[ \log {\left (y{\left (x \right )} \right )} = C_{1} + \operatorname {asin}{\left (\frac {x}{y{\left (x \right )}} \right )} \]