69.1.25 problem 42

Internal problem ID [14099]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 42
Date solved : Wednesday, March 05, 2025 at 10:32:48 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 26

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

\[ \frac {-c_{1} x^{2}+y+\sqrt {x^{2}+y^{2}}}{x^{2}} = 0 \]

✓ Mathematica. Time used: 0.291 (sec). Leaf size: 13

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

\[ y(x)\to x \sinh (\log (x)+c_1) \]

✓ Sympy. Time used: 1.144 (sec). Leaf size: 12

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

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