63.5.33 problem 16-b(ii)
Internal
problem
ID
[13036]
Book
:
A
First
Course
in
Differential
Equations
by
J.
David
Logan.
Third
Edition.
Springer-Verlag,
NY.
2015.
Section
:
Chapter
1,
First
order
differential
equations.
Section
1.4.1.
Integrating
factors.
Exercises
page
41
Problem
number
:
16-b(ii)
Date
solved
:
Monday, March 31, 2025 at 07:31:55 AM
CAS
classification
:
[_exact]
\begin{align*} t^{3}+\frac {x}{t}+\left (x^{2}+\ln \left (t \right )\right ) x^{\prime }&=0 \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 305
ode:=t^3+x(t)/t+(x(t)^2+ln(t))*diff(x(t),t) = 0;
dsolve(ode,x(t), singsol=all);
\begin{align*}
x &= \frac {\left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (t \right )}{2 \left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
x &= \frac {i \left (-\left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}-4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
x &= \frac {i \left (\left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )\right ) \sqrt {3}-\left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{2}/{3}}+4 \ln \left (t \right )}{4 \left (-3 t^{4}-12 c_1 +\sqrt {64 \ln \left (t \right )^{3}+9 \left (t^{4}+4 c_1 \right )^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 1.913 (sec). Leaf size: 307
ode=t^3+x[t]/t+(x[t]^2+Log[t])*D[x[t],t]==0;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*}
x(t)\to \frac {-4 \log (t)+\left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}}{2 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\
x(t)\to \frac {i \left (\sqrt {3}+i\right ) \left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4+4 i \sqrt {3}\right ) \log (t)}{4 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\
x(t)\to \frac {\left (-1-i \sqrt {3}\right ) \left (-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1\right ){}^{2/3}+\left (4-4 i \sqrt {3}\right ) \log (t)}{4 \sqrt [3]{-3 t^4+\sqrt {64 \log ^3(t)+9 \left (t^4-4 c_1\right ){}^2}+12 c_1}} \\
\end{align*}
✗ Sympy
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(t**3 + (x(t)**2 + log(t))*Derivative(x(t), t) + x(t)/t,0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
NotImplementedError : The given ODE Derivative(x(t), t) - (-t**4 - x(t))/(t*(x(t)**2 + log(t))) cannot be solved by the factorable group method