15.20.11 problem 11

Internal problem ID [3319]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 38, page 173
Problem number : 11
Date solved : Sunday, March 30, 2025 at 01:35:31 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} 2 {y^{\prime }}^{3} x +1&={y^{\prime }}^{2} y \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 505

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

\begin{align*} y &= -\frac {9 x^{2} 3^{{1}/{3}} \left (-\frac {2 \,3^{{2}/{3}} {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}} c_1^{2}}{9}+x \left (-\frac {2 \,3^{{1}/{3}} {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} c_1}{9}+x \right ) \left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right )\right )}{{\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} \left (c_1 3^{{1}/{3}} x +{\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}\right )^{2}} \\ y &= \frac {4 \left (3 \left (-i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) c_1^{2} {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (c_1 \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}}+9 x \right ) \left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right )\right ) x^{2} 3^{{1}/{3}}}{{\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} {\left (\left (i-\sqrt {3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x c_1 \left (3^{{5}/{6}}+i 3^{{1}/{3}}\right )\right )}^{2}} \\ y &= -\frac {4 x^{2} \left (-3 c_1^{2} \left (i 3^{{1}/{6}}+\frac {3^{{2}/{3}}}{3}\right ) {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (\left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_1 {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}}-9 x \right ) \left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right )\right ) 3^{{1}/{3}}}{{\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{1}/{3}} {\left (\left (\sqrt {3}+i\right ) {\left (\left (-9+\sqrt {\frac {-3 c_1^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}}+x \left (i 3^{{1}/{3}}-3^{{5}/{6}}\right ) c_1 \right )}^{2}} \\ \end{align*}

✓ Mathematica. Time used: 152.912 (sec). Leaf size: 17695

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

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✗ Sympy

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

Timed Out