85.1.14 problem 5 (i)
Internal
problem
ID
[22419]
Book
:
Applied
Differential
Equations.
By
Murray
R.
Spiegel.
3rd
edition.
1980.
Pearson.
ISBN
978-0130400970
Section
:
Chapter
1.
Differential
equations
in
general.
A
Exercises
at
page
12
Problem
number
:
5
(i)
Date
solved
:
Thursday, October 02, 2025 at 08:38:47 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y+\left (2 x -3 y\right ) y^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 307
ode:=y(x)+(2*x-3*y(x))*diff(y(x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {c_1 \left (-4 x^{3}+27 c_1 \right )}\right )^{{1}/{3}}}{6}+\frac {2 x^{2}}{3 \left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {c_1 \left (-4 x^{3}+27 c_1 \right )}\right )^{{1}/{3}}}+\frac {x}{3} \\
y &= \frac {i \left (-\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{2}/{3}}+4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{12 \left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{2}/{3}}-4 x^{2}\right ) \sqrt {3}-{\left (\left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}-2 x \right )}^{2}}{12 \left (-108 c_1 +8 x^{3}+12 \sqrt {3}\, \sqrt {-4 c_1 \,x^{3}+27 c_1^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 60.058 (sec). Leaf size: 379
ode=y[x]+(2*x-3*y[x])*D[y[x],{x,1}]==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \frac {1}{3} \left (\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}+\frac {x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+x\right )\\ y(x)&\to \frac {1}{12} \left (i 2^{2/3} \left (\sqrt {3}+i\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-27 e^{c_1}}-\frac {2 \left (1+i \sqrt {3}\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+4 x\right )\\ y(x)&\to \frac {1}{12} \left (-2^{2/3} \left (1+i \sqrt {3}\right ) \sqrt [3]{2 x^3+3 \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-27 e^{c_1}}+\frac {2 i \left (\sqrt {3}+i\right ) x^2}{\sqrt [3]{x^3+\frac {3}{2} \sqrt {3} \sqrt {e^{c_1} \left (-4 x^3+27 e^{c_1}\right )}-\frac {27 e^{c_1}}{2}}}+4 x\right ) \end{align*}
✓ Sympy. Time used: 27.551 (sec). Leaf size: 345
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq((2*x - 3*y(x))*Derivative(y(x), x) + y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = - \frac {\sqrt [3]{2} x^{2}}{3 \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}} + \frac {x}{3} - \frac {2^{\frac {2}{3}} \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}}{6}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}} + 2 x - 2 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}} - 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}}{6 \left (1 - \sqrt {3} i\right )}, \ y{\left (x \right )} = \frac {\frac {4 \sqrt [3]{2} x^{2}}{\sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}} + 2 x + 2 \sqrt {3} i x - 2^{\frac {2}{3}} \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}} + 2^{\frac {2}{3}} \sqrt {3} i \sqrt [3]{9 C_{1} - 2 x^{3} + 3 \sqrt {C_{1} \left (9 C_{1} - 4 x^{3}\right )}}}{6 \left (1 + \sqrt {3} i\right )}\right ]
\]