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
:
Tuesday, March 04, 2025 at 04:34:47 PM
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: 0.046 (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 \left (x \right ) &= -\frac {9 \,3^{{1}/{3}} x^{2} \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}+\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 ) x \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 \left (x \right ) &= \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 (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) \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 )\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 (\left (-9+\sqrt {\frac {-3 c_{1}^{3}+81 x}{x}}\right ) x^{2}\right )}^{{2}/{3}} \left (i-\sqrt {3}\right )+x c_{1} \left (i 3^{{1}/{3}}+3^{{5}/{6}}\right )\right )}^{2}} \\
y \left (x \right ) &= -\frac {4 x^{2} 3^{{1}/{3}} \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 (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 )}{{\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 (-3^{{5}/{6}}+i 3^{{1}/{3}}\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]
Too large to display
✗ 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