54.1.64 problem 65
Internal
problem
ID
[11378]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
linear
first
order
Problem
number
:
65
Date
solved
:
Sunday, October 12, 2025 at 01:36:46 AM
CAS
classification
:
[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]
\begin{align*} y^{\prime }-\sqrt {\frac {y^{3}+1}{x^{3}+1}}&=0 \end{align*}
✓ Maple. Time used: 0.005 (sec). Leaf size: 47
ode:=diff(y(x),x)-((y(x)^3+1)/(x^3+1))^(1/2) = 0;
dsolve(ode,y(x), singsol=all);
\[
\int _{}^{y}\frac {1}{\sqrt {\textit {\_a}^{3}+1}}d \textit {\_a} -\frac {\int _{}^{x}\sqrt {\frac {y^{3}+1}{\textit {\_a}^{3}+1}}d \textit {\_a}}{\sqrt {y^{3}+1}}+c_1 = 0
\]
✓ Mathematica. Time used: 97.356 (sec). Leaf size: 337
ode=D[y[x],x] - Sqrt[(y[x]^3+1)/(x^3+1)]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {i (\text {$\#$1}+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (\text {$\#$1}+1)}} \sqrt {1-\frac {6 i}{\left (\sqrt {3}+3 i\right ) (\text {$\#$1}+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {\text {$\#$1}+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {3 i}{2 \left (\sqrt {3}+3 i\right )}} \sqrt {\text {$\#$1}^2-\text {$\#$1}+1}}\&\right ]\left [\frac {i (x+1) \sqrt {1+\frac {6 i}{\left (\sqrt {3}-3 i\right ) (x+1)}} \sqrt {1-\frac {6 i}{\left (\sqrt {3}+3 i\right ) (x+1)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {6 i}{3 i+\sqrt {3}}}}{\sqrt {x+1}}\right ),\frac {3 i+\sqrt {3}}{3 i-\sqrt {3}}\right )}{\sqrt {-\frac {3 i}{2 \left (\sqrt {3}+3 i\right )}} \sqrt {x^2-x+1}}+c_1\right ]\\ y(x)&\to -1\\ y(x)&\to \sqrt [3]{-1}\\ y(x)&\to -(-1)^{2/3} \end{align*}
✓ Sympy. Time used: 3.343 (sec). Leaf size: 94
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-sqrt((y(x)**3 + 1)/(x**3 + 1)) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\begin {cases} \frac {y{\left (x \right )}}{\sqrt {y^{3}{\left (x \right )} + 1}} & \text {for}\: \left |{y{\left (x \right )}}\right | < 1 \\\frac {{G_{2, 2}^{1, 1}\left (\begin {matrix} 0 & 1 \\0 & -1 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {y^{3}{\left (x \right )} + 1}} + \frac {{G_{2, 2}^{0, 2}\left (\begin {matrix} 0, 1 & \\ & -1, 0 \end {matrix} \middle | {y{\left (x \right )}} \right )} y{\left (x \right )}}{\sqrt {y^{3}{\left (x \right )} + 1}} & \text {otherwise} \end {cases} - \frac {\int \sqrt {\frac {y^{3}{\left (x \right )} + 1}{x^{3} + 1}}\, dx}{\sqrt {\left (y{\left (x \right )} + 1\right ) \left (y^{2}{\left (x \right )} - y{\left (x \right )} + 1\right )}} = C_{1}
\]