22.1.57 problem 58
Internal
problem
ID
[4363]
Book
:
Differential
equations
for
engineers
by
Wei-Chau
XIE,
Cambridge
Press
2010
Section
:
Chapter
2.
First-Order
and
Simple
Higher-Order
Differential
Equations.
Page
78
Problem
number
:
58
Date
solved
:
Tuesday, September 30, 2025 at 07:23:29 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} \left (y^{3}+\frac {x}{y}\right ) y^{\prime }&=1 \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 16
ode:=(y(x)^3+x/y(x))*diff(y(x),x) = 1;
dsolve(ode,y(x), singsol=all);
\[
-y c_1 +x -\frac {y^{4}}{3} = 0
\]
✓ Mathematica. Time used: 0.077 (sec). Leaf size: 997
ode=(y[x]^3+x/y[x])*D[y[x],x]==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}}\\ y(x)&\to \frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}+\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}}\\ y(x)&\to -\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}-\frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}}\\ y(x)&\to \frac {1}{2} \sqrt {\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}-\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}+\frac {6 c_1}{\sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}}}}-\frac {1}{2} \sqrt {\frac {\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}{\sqrt [3]{2}}-\frac {4 \sqrt [3]{2} x}{\sqrt [3]{9 c_1{}^2-\sqrt {256 x^3+81 c_1{}^4}}}} \end{align*}
✓ Sympy. Time used: 1.044 (sec). Leaf size: 1017
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((x/y(x) + y(x)**3)*Derivative(y(x), x) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\text {Solution too large to show}
\]