81.6.19 problem 7-18
Internal
problem
ID
[21565]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
7.
Linear
Differential
Equations.
Page
101.
Problem
number
:
7-18
Date
solved
:
Thursday, October 02, 2025 at 07:49:35 PM
CAS
classification
:
[[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y^{2}+\left (3 y x -1\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 294
ode:=y(x)^2+(3*x*y(x)-1)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {1+{\left (6 \sqrt {6}\, \sqrt {c_1 \left (54 c_1 \,x^{2}+1\right )}\, x +108 c_1 \,x^{2}+1\right )}^{{1}/{3}}+\frac {1}{{\left (6 \sqrt {6}\, \sqrt {c_1 \left (54 c_1 \,x^{2}+1\right )}\, x +108 c_1 \,x^{2}+1\right )}^{{1}/{3}}}}{6 x} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (6 \sqrt {6}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{2}/{3}}-i \sqrt {3}-2 \left (6 \sqrt {6}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{1}/{3}}+1}{12 \left (6 \sqrt {6}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{1}/{3}} x} \\
y &= \frac {i \left (\left (6 \sqrt {3}\, \sqrt {2}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{2}/{3}}-1\right ) \sqrt {3}-{\left (\left (6 \sqrt {3}\, \sqrt {2}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{1}/{3}}-1\right )}^{2}}{12 \left (6 \sqrt {3}\, \sqrt {2}\, \sqrt {54 x^{2} c_1^{2}+c_1}\, x +108 c_1 \,x^{2}+1\right )^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 43.359 (sec). Leaf size: 331
ode=y[x]^2+(3*x*y[x]-1)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}+\frac {1}{\sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}}+1}{6 x}\\ y(x)&\to \frac {2 i \left (\sqrt {3}+i\right ) \sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}-\frac {2 \left (1+i \sqrt {3}\right )}{\sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}}+4}{24 x}\\ y(x)&\to \frac {-2 \left (1+i \sqrt {3}\right ) \sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}+\frac {2 i \left (\sqrt {3}+i\right )}{\sqrt [3]{108 c_1 x^2+6 \sqrt {6} \sqrt {c_1 x^2 \left (1+54 c_1 x^2\right )}+1}}+4}{24 x}\\ y(x)&\to 0 \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((3*x*y(x) - 1)*Derivative(y(x), x) + y(x)**2,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out