67.8.25 problem 13.4 (f)

Internal problem ID [16520]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.4 (f)
Date solved : Thursday, October 02, 2025 at 01:35:57 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Liouville, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.033 (sec). Leaf size: 293

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

\begin{align*} y &= -i \\ y &= i \\ y &= \frac {\left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{2}/{3}}-4}{2 \left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{1}/{3}}} \\ y &= -\frac {\left (1+i \sqrt {3}\right ) \left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{2}/{3}}+4 i \sqrt {3}-4}{4 \left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{1}/{3}}} \\ y &= \frac {i \sqrt {3}\, \left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{2}/{3}}+4 i \sqrt {3}-\left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{2}/{3}}+4}{4 \left (12 c_1 x +12 c_2 +4 \sqrt {9 c_1^{2} x^{2}+18 c_1 x c_2 +9 c_2^{2}+4}\right )^{{1}/{3}}} \\ \end{align*}

✓ Mathematica. Time used: 60.108 (sec). Leaf size: 307

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

\begin{align*} y(x)&\to \frac {-2+\sqrt [3]{2} \left (3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1\right ){}^{2/3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}}+\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}\\ y(x)&\to \frac {1-i \sqrt {3}}{2^{2/3} \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{3 c_1 x+\sqrt {4+9 c_1{}^2 (x+c_2){}^2}+3 c_2 c_1}}{2 \sqrt [3]{2}} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt(-(y(x) + 1/y(x))*Derivative(y(x), (x, 2)))/2 + Derivative(y(x), x) cannot be solved by the factorable group method