80.4.4 problem 4
Internal
problem
ID
[18476]
Book
:
Elementary
Differential
Equations.
By
Thornton
C.
Fry.
D
Van
Nostrand.
NY.
First
Edition
(1929)
Section
:
Chapter
IV.
Methods
of
solution:
First
order
equations.
section
31.
Problems
at
page
85
Problem
number
:
4
Date
solved
:
Thursday, March 13, 2025 at 12:06:06 PM
CAS
classification
:
[_linear]
\begin{align*} p^{\prime }&=\frac {p+a \,t^{3}-2 p t^{2}}{t \left (-t^{2}+1\right )} \end{align*}
✓ Maple. Time used: 0.001 (sec). Leaf size: 20
ode:=diff(p(t),t) = (p(t)+a*t^3-2*p(t)*t^2)/t/(-t^2+1);
dsolve(ode,p(t), singsol=all);
\[
p = t \left (\sqrt {t +1}\, \sqrt {t -1}\, c_{1} +a \right )
\]
✓ Mathematica. Time used: 0.048 (sec). Leaf size: 23
ode=D[p[t],t]==(p[t]+a*t^3-2*p[t]*t^2 )/(t*(1-t^2));
ic={};
DSolve[{ode,ic},p[t],t,IncludeSingularSolutions->True]
\[
p(t)\to t \left (a+c_1 \sqrt {1-t^2}\right )
\]
✓ Sympy. Time used: 36.674 (sec). Leaf size: 338
from sympy import *
t = symbols("t")
a = symbols("a")
p = Function("p")
ode = Eq(Derivative(p(t), t) - (a*t**3 - 2*t**2*p(t) + p(t))/(t*(1 - t**2)),0)
ics = {}
dsolve(ode,func=p(t),ics=ics)
\[
p{\left (t \right )} = \begin {cases} \frac {C_{1} \sqrt {1 - t^{2}} \sqrt {t^{2} - 1}}{2 t \sqrt {1 - t^{2}} + 2 i t \sqrt {t^{2} - 1} - 2 \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {t}{\sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\\frac {i t}{\sqrt {1 - t^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {2 t}{\sqrt {t^{2} - 1}} + \frac {1}{t \sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\- \frac {2 i t^{2} \sqrt {1 - t^{2}}}{t^{3} - t} + \frac {i \sqrt {1 - t^{2}}}{t^{3} - t} & \text {otherwise} \end {cases}\right )} + \frac {a \sqrt {1 - t^{2}}}{2 t \sqrt {1 - t^{2}} + 2 i t \sqrt {t^{2} - 1} - 2 \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {t}{\sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\\frac {i t}{\sqrt {1 - t^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {2 t}{\sqrt {t^{2} - 1}} + \frac {1}{t \sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\- \frac {2 i t^{2} \sqrt {1 - t^{2}}}{t^{3} - t} + \frac {i \sqrt {1 - t^{2}}}{t^{3} - t} & \text {otherwise} \end {cases}\right )} & \text {for}\: t > -1 \wedge t < 1 \\\text {NaN} & \text {otherwise} \end {cases}
\]