74.4.4 problem 4

Internal problem ID [19841]
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, October 02, 2025 at 04:48:09 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: 0.002 (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);
 

\[ p = t \left (c_1 \sqrt {t +1}\, \sqrt {t -1}+a \right ) \]

✓ Mathematica. Time used: 0.032 (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]
 

\begin{align*} p(t)&\to t \left (a+c_1 \sqrt {1-t^2}\right ) \end{align*}

✓ Sympy. Time used: 24.190 (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)
 

\[ p{\left (t \right )} = \begin {cases} \frac {C_{1} \sqrt {1 - t^{2}} \sqrt {t^{2} - 1}}{2 t \sqrt {1 - t^{2}} + 2 i t \sqrt {t^{2} - 1} - 2 \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {t}{\sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\\frac {i t}{\sqrt {1 - t^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {2 t}{\sqrt {t^{2} - 1}} + \frac {1}{t \sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\- \frac {2 i t^{2} \sqrt {1 - t^{2}}}{t^{3} - t} + \frac {i \sqrt {1 - t^{2}}}{t^{3} - t} & \text {otherwise} \end {cases}\right )} + \frac {a \sqrt {1 - t^{2}}}{2 t \sqrt {1 - t^{2}} + 2 i t \sqrt {t^{2} - 1} - 2 \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {t}{\sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\\frac {i t}{\sqrt {1 - t^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {1 - t^{2}} \sqrt {t^{2} - 1} \left (\begin {cases} - \frac {2 t}{\sqrt {t^{2} - 1}} + \frac {1}{t \sqrt {t^{2} - 1}} & \text {for}\: \left |{t^{2}}\right | > 1 \\- \frac {2 i t^{2} \sqrt {1 - t^{2}}}{t^{3} - t} + \frac {i \sqrt {1 - t^{2}}}{t^{3} - t} & \text {otherwise} \end {cases}\right )} & \text {for}\: t > -1 \wedge t < 1 \\\text {NaN} & \text {otherwise} \end {cases} \]