23.4.68 problem 68

Internal problem ID [6370]
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 : 68
Date solved : Tuesday, September 30, 2025 at 02:53:32 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{3} y^{\prime }+y^{\prime \prime }&=y y^{\prime } \sqrt {y^{4}+4 y^{\prime }} \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 133

ode:=y(x)^3*diff(y(x),x)+diff(diff(y(x),x),x) = y(x)*diff(y(x),x)*(y(x)^4+4*diff(y(x),x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {2^{{2}/{3}} \left (\left (4 c_1 +3 x \right )^{2}\right )^{{1}/{3}}}{4 c_1 +3 x} \\ y &= -\frac {2^{{2}/{3}} \left (\left (4 c_1 +3 x \right )^{2}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{8 c_1 +6 x} \\ y &= \frac {2^{{2}/{3}} \left (\left (4 c_1 +3 x \right )^{2}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{8 c_1 +6 x} \\ y &= \tan \left (\frac {c_2 +x}{c_1^{3}}\right ) \sqrt {\frac {1}{c_1^{2}}} \\ y &= \tanh \left (\frac {c_2 +x}{c_1^{3}}\right ) \sqrt {\frac {1}{c_1^{2}}} \\ \end{align*}

✓ Mathematica. Time used: 60.529 (sec). Leaf size: 95

ode=y[x]^3*D[y[x],x] + D[y[x],{x,2}] == y[x]*D[y[x],x]*Sqrt[y[x]^4 + 4*D[y[x],x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (-1)^{3/4} \sqrt {(\cosh (c_1)+\sinh (c_1)){}^2} \tan \left (\sqrt [4]{-1} (x+c_2) \left ((\cosh (c_1)+\sinh (c_1)){}^2\right ){}^{3/2}\right )\\ y(x)&\to \sqrt [4]{-1} \sqrt {(\cosh (c_1)+\sinh (c_1)){}^2} \tan \left ((-1)^{3/4} (x+c_2) \left ((\cosh (c_1)+\sinh (c_1)){}^2\right ){}^{3/2}\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-sqrt(y(x)**4 + 4*Derivative(y(x), x))*y(x)*Derivative(y(x), x) + y(x)**3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE (sqrt(-27*y(x)**3*Derivative(y(x), (x, 2))**3/2 + 729*Derivative