63.1.36 problem 55

Internal problem ID [15476]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 55
Date solved : Thursday, October 02, 2025 at 10:18:08 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.030 (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);
 

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

✓ Mathematica. Time used: 0.225 (sec). Leaf size: 27

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

\begin{align*} y(x)&\to \frac {1}{2} e^{-c_1} \left (x^2-e^{2 c_1}\right ) \end{align*}

✓ Sympy. Time used: 46.588 (sec). Leaf size: 122

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)
 

\[ \left [ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {- 4 x^{4} e^{- C_{1}} - 4 x^{2} - e^{C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {- 4 x^{4} e^{- C_{1}} - 4 x^{2} - e^{C_{1}}}}{4}, \ y{\left (x \right )} = - \frac {\sqrt {2} \sqrt {4 x^{4} e^{- C_{1}} - 4 x^{2} + e^{C_{1}}}}{4}, \ y{\left (x \right )} = \frac {\sqrt {2} \sqrt {4 x^{4} e^{- C_{1}} - 4 x^{2} + e^{C_{1}}}}{4}\right ] \]