15.2.21 problem 21

Internal problem ID [2891]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 6, page 25
Problem number : 21
Date solved : Sunday, March 30, 2025 at 12:43:49 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.120 (sec). Leaf size: 32

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

\[ -c_1 +\sqrt {x^{2}+y^{2}}\, y^{k -1}+x y^{k -1} = 0 \]

✓ Mathematica. Time used: 0.248 (sec). Leaf size: 59

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

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

✓ Sympy. Time used: 2.550 (sec). Leaf size: 14

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

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