23.2.1 problem 1

Internal problem ID [5356]
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 : 1
Date solved : Tuesday, September 30, 2025 at 12:36:04 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}&=a \,x^{n} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 51

ode:=diff(y(x),x)^2 = a*x^n; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 57

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

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

✓ Sympy. Time used: 0.298 (sec). Leaf size: 54

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

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