89.3.6 problem 6
Internal
problem
ID
[24304]
Book
:
A
short
course
in
Differential
Equations.
Earl
D.
Rainville.
Second
edition.
1958.
Macmillan
Publisher,
NY.
CAT
58-5010
Section
:
Chapter
2.
Equations
of
the
first
order
and
first
degree.
Exercises
at
page
34
Problem
number
:
6
Date
solved
:
Thursday, October 02, 2025 at 10:12:46 PM
CAS
classification
:
[_exact, _rational]
\begin{align*} v \left (2 u v^{2}-3\right )+\left (3 u^{2} v^{2}-3 u +4 v\right ) v^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 630
ode:=v(u)*(2*u*v(u)^2-3)+(3*u^2*v(u)^2-3*u+4*v(u))*diff(v(u),u) = 0;
dsolve(ode,v(u), singsol=all);
\begin{align*}
v &= \frac {-2+\frac {\left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}}{2}+\frac {18 u^{3}+8}{\left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}}}{3 u^{2}} \\
v &= \frac {36 i \sqrt {3}\, u^{3}-i \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{2}/{3}} \sqrt {3}-36 u^{3}+16 i \sqrt {3}-\left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{2}/{3}}-8 \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}-16}{12 u^{2} \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}} \\
v &= -\frac {36 i \sqrt {3}\, u^{3}-i \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{2}/{3}} \sqrt {3}+36 u^{3}+16 i \sqrt {3}+\left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{2}/{3}}+8 \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}+16}{12 u^{2} \left (-108 c_1 \,u^{4}+12 \sqrt {3}\, \sqrt {27 c_1^{2} u^{4}-108 u^{5}+108 c_1 \,u^{3}-36 u^{2}+32 c_1}\, u^{2}-216 u^{3}-64\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 35.058 (sec). Leaf size: 474
ode=v[u]*(2*u*v[u]^2-3)+( 3*u^2*v[u]^2-3*u+4*v[u])*D[v[u],u]==0;
ic={};
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
\begin{align*} v(u)&\to -\frac {2}{3 u^2}-\frac {\sqrt [3]{2} \left (-9 u^3-4\right )}{3 u^2 \sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}+\frac {\sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}{3 \sqrt [3]{2} u^2}\\ v(u)&\to -\frac {2}{3 u^2}+\frac {\left (1+i \sqrt {3}\right ) \left (-9 u^3-4\right )}{3\ 2^{2/3} u^2 \sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}{6 \sqrt [3]{2} u^2}\\ v(u)&\to -\frac {2}{3 u^2}+\frac {\left (1-i \sqrt {3}\right ) \left (-9 u^3-4\right )}{3\ 2^{2/3} u^2 \sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{27 c_1 u^4-54 u^3+\sqrt {-4 \left (9 u^3+4\right )^3+\left (-27 c_1 u^4+54 u^3+16\right ){}^2}-16}}{6 \sqrt [3]{2} u^2} \end{align*}
✗ Sympy
from sympy import *
u = symbols("u")
v = Function("v")
ode = Eq((2*u*v(u)**2 - 3)*v(u) + (3*u**2*v(u)**2 - 3*u + 4*v(u))*Derivative(v(u), u),0)
ics = {}
dsolve(ode,func=v(u),ics=ics)
Timed Out