84.8.5 problem 4.14
Internal
problem
ID
[22117]
Book
:
Schaums
outline
series.
Differential
Equations
By
Richard
Bronson.
1973.
McGraw-Hill
Inc.
ISBN
0-07-008009-7
Section
:
Chapter
4.
Separable
first-order
differential
equations.
Supplementary
problems
Problem
number
:
4.14
Date
solved
:
Thursday, October 02, 2025 at 08:25:25 PM
CAS
classification
:
[_separable]
\begin{align*} x^{2}+1+\left (y^{2}+y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 444
ode:=x^2+1+(y(x)^2+y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4 x^{6}+24 c_1 \,x^{3}+24 x^{4}+2 x^{3}+36 c_1^{2}+72 c_1 x +36 x^{2}+6 c_1 +6 x}\right )^{{1}/{3}}}{2}+\frac {1}{2 \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4 x^{6}+24 c_1 \,x^{3}+24 x^{4}+2 x^{3}+36 c_1^{2}+72 c_1 x +36 x^{2}+6 c_1 +6 x}\right )^{{1}/{3}}}-\frac {1}{2} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{2}/{3}}-i \sqrt {3}+2 \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{1}/{3}}+1}{4 \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{1}/{3}}} \\
y &= \frac {\left (i \sqrt {3}-1\right ) \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{2}/{3}}-i \sqrt {3}-2 \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{1}/{3}}-1}{4 \left (-1-4 x^{3}-12 c_1 -12 x +2 \sqrt {4}\, \sqrt {\left (x^{3}+3 c_1 +3 x \right ) \left (x^{3}+3 x +3 c_1 +\frac {1}{2}\right )}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.393 (sec). Leaf size: 323
ode=(x^2+1)+(y[x]^2+y[x])*D[y[x],x]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{2} \left (\sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}+\frac {1}{\sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}}-1\right )\\ y(x)&\to \frac {1}{4} \left (i \left (\sqrt {3}+i\right ) \sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}-\frac {1+i \sqrt {3}}{\sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}}-2\right )\\ y(x)&\to \frac {1}{4} \left (-\left (\left (1+i \sqrt {3}\right ) \sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}\right )+\frac {i \left (\sqrt {3}+i\right )}{\sqrt [3]{-4 x^3+\sqrt {-1+\left (4 x^3+12 x+1-12 c_1\right ){}^2}-12 x-1+12 c_1}}-2\right ) \end{align*}
✓ Sympy. Time used: 75.792 (sec). Leaf size: 636
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2 + (y(x)**2 + y(x))*Derivative(y(x), x) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}{12} + \frac {\sqrt {3} i \sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}{12} - \frac {1}{2} - \frac {3}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}, \ y{\left (x \right )} = \frac {\sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}{12} - \frac {\sqrt {3} i \sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}{12} - \frac {1}{2} - \frac {3}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}, \ y{\left (x \right )} = - \frac {\sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}{6} - \frac {1}{2} - \frac {3}{2 \sqrt [3]{- 324 C_{1} + 108 x^{3} + 324 x + 4 \sqrt {6561 C_{1}^{2} - 4374 C_{1} x^{3} - 13122 C_{1} x - \frac {2187 C_{1}}{2} + 729 x^{6} + 4374 x^{4} + \frac {729 x^{3}}{2} + 6561 x^{2} + \frac {2187 x}{2}} + 27}}\right ]
\]