81.1.15 problem 2-14 (a)
Internal
problem
ID
[21460]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
2.
Separable
differential
equations
Problem
number
:
2-14
(a)
Date
solved
:
Thursday, October 02, 2025 at 07:38:40 PM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {1+x}{1+y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 376
ode:=diff(y(x),x) = (1+x)/(1+y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{2}/{3}}-4}{2 \left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{2}/{3}}+4 i \sqrt {3}-4}{4 \left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {4+i \left (\left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{2}/{3}}+4\right ) \sqrt {3}-\left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{2}/{3}}}{4 \left (6 x^{2}+12 c_1 +12 x +2 \sqrt {16+9 x^{4}+36 x^{3}+36 \left (1+c_1 \right ) x^{2}+72 c_1 x +36 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.063 (sec). Leaf size: 335
ode=D[y[x],x]==(x+1)/( y[x]^2+1 );
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {-2\ 2^{2/3}+\sqrt [3]{2} \left (3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1\right ){}^{2/3}}{2 \sqrt [3]{3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1}}\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1}}{2\ 2^{2/3}}+\frac {1+i \sqrt {3}}{\sqrt [3]{2} \sqrt [3]{3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1}}\\ y(x)&\to \frac {1-i \sqrt {3}}{\sqrt [3]{2} \sqrt [3]{3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1}}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{3 x^2+\sqrt {16+9 \left (x^2+2 x+2 c_1\right ){}^2}+6 x+6 c_1}}{2\ 2^{2/3}} \end{align*}
✓ Sympy. Time used: 104.925 (sec). Leaf size: 530
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-x - 1)/(y(x)**2 + 1) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}{12} + \frac {\sqrt {3} i \sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}{12} + \frac {6 \sqrt [3]{2}}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}{12} - \frac {\sqrt {3} i \sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}{12} + \frac {6 \sqrt [3]{2}}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}{6} + \frac {3 \sqrt [3]{2}}{\sqrt [3]{- 162 C_{1} - 81 x^{2} - 162 x + 2 \sqrt {6561 C_{1}^{2} + 6561 C_{1} x^{2} + 13122 C_{1} x + \frac {6561 x^{4}}{4} + 6561 x^{3} + 6561 x^{2} + 2916}}}\right )\right ]
\]