90.3.13 problem 13
Internal
problem
ID
[25077]
Book
:
Ordinary
Differential
Equations.
By
William
Adkins
and
Mark
G
Davidson.
Springer.
NY.
2010.
ISBN
978-1-4614-3617-1
Section
:
Chapter
1.
First
order
differential
equations.
Exercises
at
page
41
Problem
number
:
13
Date
solved
:
Thursday, October 02, 2025 at 11:49:26 PM
CAS
classification
:
[_separable]
\begin{align*} y^{4} y^{\prime }&=t +2 \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 176
ode:=y(t)^4*diff(y(t),t) = t+2;
dsolve(ode,y(t), singsol=all);
\begin{align*}
y &= \frac {\left (80 t^{2}+32 c_1 +320 t \right )^{{1}/{5}}}{2} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}+\sqrt {5}+1\right ) \left (80 t^{2}+32 c_1 +320 t \right )^{{1}/{5}}}{8} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5-\sqrt {5}}-\sqrt {5}-1\right ) \left (80 t^{2}+32 c_1 +320 t \right )^{{1}/{5}}}{8} \\
y &= -\frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}-\sqrt {5}+1\right ) \left (80 t^{2}+32 c_1 +320 t \right )^{{1}/{5}}}{8} \\
y &= \frac {\left (i \sqrt {2}\, \sqrt {5+\sqrt {5}}+\sqrt {5}-1\right ) \left (80 t^{2}+32 c_1 +320 t \right )^{{1}/{5}}}{8} \\
\end{align*}
✓ Mathematica. Time used: 0.124 (sec). Leaf size: 153
ode=y[t]^4*D[y[t],{t,1}]==t+2;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to -\sqrt [5]{-\frac {5}{2}} \sqrt [5]{t^2+4 t+2 c_1}\\ y(t)&\to \sqrt [5]{\frac {5}{2}} \sqrt [5]{t^2+4 t+2 c_1}\\ y(t)&\to (-1)^{2/5} \sqrt [5]{\frac {5}{2}} \sqrt [5]{t^2+4 t+2 c_1}\\ y(t)&\to -(-1)^{3/5} \sqrt [5]{\frac {5}{2}} \sqrt [5]{t^2+4 t+2 c_1}\\ y(t)&\to (-1)^{4/5} \sqrt [5]{\frac {5}{2}} \sqrt [5]{t^2+4 t+2 c_1} \end{align*}
✓ Sympy. Time used: 5.563 (sec). Leaf size: 189
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-t + y(t)**4*Derivative(y(t), t) - 2,0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
\left [ y{\left (t \right )} = \sqrt [5]{C_{1} + \frac {5 t^{2}}{2} + 10 t}, \ y{\left (t \right )} = \frac {\left (- \sqrt {5} - 1 - \sqrt {2} i \sqrt {5 - \sqrt {5}}\right ) \sqrt [5]{C_{1} + \frac {5 t^{2}}{2} + 10 t}}{4}, \ y{\left (t \right )} = \frac {\left (- \sqrt {5} - 1 + \sqrt {2} i \sqrt {5 - \sqrt {5}}\right ) \sqrt [5]{C_{1} + \frac {5 t^{2}}{2} + 10 t}}{4}, \ y{\left (t \right )} = \frac {\left (-1 + \sqrt {5} - \sqrt {2} i \sqrt {\sqrt {5} + 5}\right ) \sqrt [5]{C_{1} + \frac {5 t^{2}}{2} + 10 t}}{4}, \ y{\left (t \right )} = \frac {\left (-1 + \sqrt {5} + \sqrt {2} i \sqrt {\sqrt {5} + 5}\right ) \sqrt [5]{C_{1} + \frac {5 t^{2}}{2} + 10 t}}{4}\right ]
\]