56.1.3 problem Ex 3

Internal problem ID [14081]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter 2, differential equations of the first order and the first degree. Article 8. Exact differential equations. Page 11
Problem number : Ex 3
Date solved : Thursday, October 02, 2025 at 09:10:53 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _dAlembert]

\begin{align*} \frac {1}{\sqrt {x^{2}+y^{2}}}+\left (\frac {1}{y}-\frac {x}{y \sqrt {x^{2}+y^{2}}}\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 18

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

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

✓ Mathematica. Time used: 0.285 (sec). Leaf size: 62

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

\begin{align*} y(x)&\to -e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to e^{\frac {c_1}{2}} \sqrt {-2 x+e^{c_1}}\\ y(x)&\to 0\\ y(x)&\to \text {ComplexInfinity} \end{align*}

✓ Sympy. Time used: 1.459 (sec). Leaf size: 12

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

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