23.2.73 problem 75

Internal problem ID [5428]
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 : 75
Date solved : Tuesday, September 30, 2025 at 12:42:05 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-\left (4+y^{2}\right ) y^{\prime }+4+y^{2}&=0 \end{align*}

✓ Maple. Time used: 0.042 (sec). Leaf size: 75

ode:=diff(y(x),x)^2-(4+y(x)^2)*diff(y(x),x)+4+y(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -2 i \\ y &= 2 i \\ x +2 \int _{}^{y}\frac {1}{-\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}-4}d \textit {\_a} -c_1 &= 0 \\ x -2 \int _{}^{y}\frac {1}{\textit {\_a}^{2}+\sqrt {\textit {\_a}^{2} \left (\textit {\_a}^{2}+4\right )}+4}d \textit {\_a} -c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.328 (sec). Leaf size: 73

ode=(D[y[x],x])^2-(4+y[x]^2)*D[y[x],x]+4+y[x]^2==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2-4 c_1 x-1+4 c_1{}^2}{x-2 c_1}\\ y(x)&\to \frac {x^2+4 c_1 x-1+4 c_1{}^2}{x+2 c_1}\\ y(x)&\to -2 i\\ y(x)&\to 2 i \end{align*}

✓ Sympy. Time used: 2.399 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-y(x)**2 - 4)*Derivative(y(x), x) + y(x)**2 + Derivative(y(x), x)**2 + 4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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