23.4.64 problem 64

Internal problem ID [6366]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 64
Date solved : Tuesday, September 30, 2025 at 02:53:19 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=a x \left (1+{y^{\prime }}^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 57

ode:=diff(diff(y(x),x),x) = a*x*(1+diff(y(x),x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= a \int \sqrt {-\frac {1}{-4+a^{2} \left (x^{2}+2 c_1 \right )^{2}}}\, \left (x^{2}+2 c_1 \right )d x +c_2 \\ \end{align*}

✓ Mathematica. Time used: 62.683 (sec). Leaf size: 332

ode=D[y[x],{x,2}] == a*x*(1 + D[y[x],x]^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-\frac {\sqrt {\frac {a x^2-2+2 c_1}{-1+c_1}} \sqrt {\frac {a x^2+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{2 c_1+2}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {a^2 x^4+4 a c_1 x^2-4+4 c_1{}^2}}\\ y(x)&\to \frac {\sqrt {\frac {a x^2-2+2 c_1}{-1+c_1}} \sqrt {\frac {a x^2+2+2 c_1}{1+c_1}} \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{2 c_1+2}}\right ),\frac {c_1+1}{c_1-1}\right )+(-1+c_1) E\left (i \text {arcsinh}\left (x \sqrt {\frac {a}{2 c_1+2}}\right )|\frac {c_1+1}{c_1-1}\right )\right )}{\sqrt {\frac {a}{2+2 c_1}} \sqrt {a^2 x^4+4 a c_1 x^2-4+4 c_1{}^2}}+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x*(Derivative(y(x), x)**2 + 1)**(3/2) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-(Derivative(y(x), (x, 2))**2/(a**2*x**2))**(1/3)/2 + sqrt