84.12.2 problem 6.7
Internal
problem
ID
[22144]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
6.
Exact
first-order
differential
equations.
Supplementary
problems
Problem
number
:
6.7
Date
solved
:
Thursday, October 02, 2025 at 08:32:23 PM
CAS
classification
:
[_exact, _rational, [_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} y+2 x y^{3}+\left (1+3 x^{2} y^{2}+x \right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 285
ode:=y(x)+2*x*y(x)^3+(1+3*x^2*y(x)^2+x)*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{2}/{3}}-12 x -12}{6 x \left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{1}/{3}}} \\
y &= -\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{12}\right ) \left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{2}/{3}}+\left (i \sqrt {3}-1\right ) \left (x +1\right )}{\left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{1}/{3}} x} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{2}/{3}}+12 \left (x +1\right ) \left (1+i \sqrt {3}\right )}{12 \left (12 \sqrt {3}\, \sqrt {4+4 x^{3}+\left (27 c_1^{2}+12\right ) x^{2}+12 x}-108 c_1 x \right )^{{1}/{3}} x} \\
\end{align*}
✓ Mathematica. Time used: 33.583 (sec). Leaf size: 340
ode=(y[x]+2*x*y[x]^3)+(1+3*x^2*y[x]^2+x)*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}{3 \sqrt [3]{2} x^2}-\frac {\sqrt [3]{2} (x+1)}{\sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}\\ y(x)&\to \frac {\left (1+i \sqrt {3}\right ) (x+1)}{2^{2/3} \sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2}\\ y(x)&\to \frac {\left (1-i \sqrt {3}\right ) (x+1)}{2^{2/3} \sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 x^4+\sqrt {108 (x+1)^3 x^6+729 c_1{}^2 x^8}}}{6 \sqrt [3]{2} x^2} \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(2*x*y(x)**3 + (3*x**2*y(x)**2 + x + 1)*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out