1.5.51 problem 51

Internal problem ID [155]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 1. First order differential equations. Section 1.6 (substitution and exact equations). Problems at page 72
Problem number : 51
Date solved : Tuesday, September 30, 2025 at 03:47:23 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }&=2 y {y^{\prime }}^{3} \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 320

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

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

✓ Mathematica. Time used: 5.938 (sec). Leaf size: 351

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

\begin{align*} y(x)&\to \frac {\sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}-\frac {\sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}{\sqrt [3]{2}}\\ y(x)&\to \frac {2^{2/3} \left (1-i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+\sqrt [3]{2} \left (-2-2 i \sqrt {3}\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}\\ y(x)&\to \frac {2^{2/3} \left (1+i \sqrt {3}\right ) \left (\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2\right ){}^{2/3}+2 i \sqrt [3]{2} \left (\sqrt {3}+i\right ) c_1}{4 \sqrt [3]{\sqrt {9 x^2+18 c_2 x+4 c_1{}^3+9 c_2{}^2}+3 x+3 c_2}}\\ y(x)&\to 0 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE 2**(2/3)*(Derivative(y(x), (x, 2))/y(x))**(1/3)/4 - 2**(2/3)*sqrt(3)*I*(Derivative(y(x), (x, 2))/y(x))**(1/3)/4 + Derivative(y(x), x) cannot be solved by the factorable group method