68.7.22 problem 22
Internal
problem
ID
[17386]
Book
:
INTRODUCTORY
DIFFERENTIAL
EQUATIONS.
Martha
L.
Abell,
James
P.
Braselton.
Fourth
edition
2014.
ElScAe.
2014
Section
:
Chapter
2.
First
Order
Equations.
Exercises
2.5,
page
64
Problem
number
:
22
Date
solved
:
Thursday, October 02, 2025 at 02:15:45 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]
\begin{align*} y^{\prime }&=\frac {t +4 y}{4 t +y} \end{align*}
✓ Maple. Time used: 0.032 (sec). Leaf size: 185
ode:=diff(y(t),t) = (t+4*y(t))/(4*t+y(t));
dsolve(ode,y(t), singsol=all);
\[
y = t \left ({\mathrm e}^{\frac {2 c_1}{3}} {\left (\frac {\left (12 \,{\mathrm e}^{2 c_1} t^{2} \operatorname {RootOf}\left (\textit {\_Z}^{5} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}-\textit {\_Z}^{3}-2\right )^{3}+8 \,{\mathrm e}^{c_1} t \operatorname {RootOf}\left (\textit {\_Z}^{3}-t \,{\mathrm e}^{c_1}, \operatorname {index} =1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{5} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}-\textit {\_Z}^{3}-2\right )^{4}+8 t^{2} {\mathrm e}^{2 c_1}+12 \operatorname {RootOf}\left (\textit {\_Z}^{5} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}-\textit {\_Z}^{3}-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{3}-t \,{\mathrm e}^{c_1}, \operatorname {index} =1\right ) {\mathrm e}^{c_1} t +2 \operatorname {RootOf}\left (\textit {\_Z}^{5} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}-\textit {\_Z}^{3}-2\right )^{2} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{5} \left (t \,{\mathrm e}^{c_1}\right )^{{2}/{3}}-\textit {\_Z}^{3}-2\right )^{3}+2\right ) {\mathrm e}^{-4 c_1}}{t^{2}}\right )}^{{1}/{3}}-1\right )
\]
✓ Mathematica. Time used: 6.718 (sec). Leaf size: 456
ode=D[y[t],t]==(t+4*y[t])/(4*t+y[t]);
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \text {Root}\left [\text {$\#$1}^5-5 \text {$\#$1}^4 t+\text {$\#$1}^3 \left (10 t^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (-10 t^3+3 e^{2 c_1} t\right )+\text {$\#$1} \left (5 t^4+3 e^{2 c_1} t^2\right )-t^5+e^{2 c_1} t^3\&,1\right ]\\ y(t)&\to \text {Root}\left [\text {$\#$1}^5-5 \text {$\#$1}^4 t+\text {$\#$1}^3 \left (10 t^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (-10 t^3+3 e^{2 c_1} t\right )+\text {$\#$1} \left (5 t^4+3 e^{2 c_1} t^2\right )-t^5+e^{2 c_1} t^3\&,2\right ]\\ y(t)&\to \text {Root}\left [\text {$\#$1}^5-5 \text {$\#$1}^4 t+\text {$\#$1}^3 \left (10 t^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (-10 t^3+3 e^{2 c_1} t\right )+\text {$\#$1} \left (5 t^4+3 e^{2 c_1} t^2\right )-t^5+e^{2 c_1} t^3\&,3\right ]\\ y(t)&\to \text {Root}\left [\text {$\#$1}^5-5 \text {$\#$1}^4 t+\text {$\#$1}^3 \left (10 t^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (-10 t^3+3 e^{2 c_1} t\right )+\text {$\#$1} \left (5 t^4+3 e^{2 c_1} t^2\right )-t^5+e^{2 c_1} t^3\&,4\right ]\\ y(t)&\to \text {Root}\left [\text {$\#$1}^5-5 \text {$\#$1}^4 t+\text {$\#$1}^3 \left (10 t^2+e^{2 c_1}\right )+\text {$\#$1}^2 \left (-10 t^3+3 e^{2 c_1} t\right )+\text {$\#$1} \left (5 t^4+3 e^{2 c_1} t^2\right )-t^5+e^{2 c_1} t^3\&,5\right ] \end{align*}
✓ Sympy. Time used: 0.614 (sec). Leaf size: 26
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq((-t - 4*y(t))/(4*t + y(t)) + Derivative(y(t), t),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\log {\left (t \right )} = C_{1} + \log {\left (\frac {\left (1 + \frac {y{\left (t \right )}}{t}\right )^{\frac {3}{2}}}{\left (-1 + \frac {y{\left (t \right )}}{t}\right )^{\frac {5}{2}}} \right )}
\]