30.9.1 problem 8

Internal problem ID [7579]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Review problem. (I) Solar Collector. page 87
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 04:54:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 27

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

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

✓ Mathematica. Time used: 0.273 (sec). Leaf size: 57

ode=D[y[x],x]==( - x + Sqrt[x^2+y[x]^2] )/y[x]; 
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 \end{align*}

✓ Sympy. Time used: 48.057 (sec). Leaf size: 235

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

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