23.2.32 problem 33

Internal problem ID [5387]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 33
Date solved : Tuesday, September 30, 2025 at 12:39:43 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}+x y^{\prime }+1&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 63

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

\begin{align*} y &= -\frac {x^{2}}{4}-\frac {x \sqrt {x^{2}-4}}{4}+\ln \left (x +\sqrt {x^{2}-4}\right )+c_1 \\ y &= \frac {x \sqrt {x^{2}-4}}{4}-\ln \left (x +\sqrt {x^{2}-4}\right )-\frac {x^{2}}{4}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.03 (sec). Leaf size: 83

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

\begin{align*} y(x)&\to -\text {arctanh}\left (\frac {x}{\sqrt {x^2-4}}\right )-\frac {x^2}{4}+\frac {1}{4} \sqrt {x^2-4} x+c_1\\ y(x)&\to \text {arctanh}\left (\frac {x}{\sqrt {x^2-4}}\right )-\frac {x^2}{4}-\frac {1}{4} \sqrt {x^2-4} x+c_1 \end{align*}

✓ Sympy. Time used: 0.202 (sec). Leaf size: 65

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

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