81.15.4 problem 19-4
Internal
problem
ID
[21712]
Book
:
The
Differential
Equations
Problem
Solver.
VOL.
I.
M.
Fogiel
director.
REA,
NY.
1978.
ISBN
78-63609
Section
:
Chapter
19.
Change
of
variables.
Page
483
Problem
number
:
19-4
Date
solved
:
Thursday, October 02, 2025 at 08:00:37 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {2 x +y}{y} \end{align*}
✓ Maple. Time used: 0.020 (sec). Leaf size: 790
ode:=diff(y(x),x) = (y(x)+2*x)/y(x);
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 24.56 (sec). Leaf size: 486
ode=D[y[x],x]==(2*x+y[x])/y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {\sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}+x\\ y(x)&\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}{2 \sqrt [3]{2}}-\frac {\left (1+i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}+x\\ y(x)&\to -\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}{2 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) x^2}{2^{2/3} \sqrt [3]{-2 x^3+\sqrt {e^{6 c_1}-4 e^{3 c_1} x^3}+e^{3 c_1}}}+x\\ y(x)&\to \sqrt [3]{-x^3}+\frac {x^2}{\sqrt [3]{-x^3}}+x\\ y(x)&\to -\frac {1}{2} i \left (\sqrt {3}-i\right ) \sqrt [3]{-x^3}+\frac {\left (1-i \sqrt {3}\right ) \left (-x^3\right )^{2/3}}{2 x}+x\\ y(x)&\to \frac {1}{2} i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^3}+\frac {\left (1+i \sqrt {3}\right ) \left (-x^3\right )^{2/3}}{2 x}+x \end{align*}
✓ Sympy. Time used: 16.859 (sec). Leaf size: 248
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((-2*x - y(x))/y(x) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \frac {\frac {2 x^{2}}{\sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} + x - \sqrt {3} i x - \sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}} - \sqrt {3} i \sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}}{1 - \sqrt {3} i}, \ y{\left (x \right )} = \frac {\frac {2 x^{2}}{\sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} + x + \sqrt {3} i x - \sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}} + \sqrt {3} i \sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}}{1 + \sqrt {3} i}, \ y{\left (x \right )} = - \frac {x^{2}}{\sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}} + x - \sqrt [3]{- 2 C_{1} + x^{3} + 2 \sqrt {C_{1} \left (C_{1} - x^{3}\right )}}\right ]
\]