89.1.26 problem 26

Internal problem ID [24261]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 2. Equations of the first order and first degree. Exercises at page 21
Problem number : 26
Date solved : Thursday, October 02, 2025 at 10:02:19 PM
CAS classification : [_separable]

\begin{align*} r^{\prime }&=-2 r t \end{align*}

With initial conditions

\begin{align*} r \left (0\right )&=r_{0} \\ \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 12

ode:=diff(r(t),t) = -2*r(t)*t; 
ic:=[r(0) = r__0]; 
dsolve([ode,op(ic)],r(t), singsol=all);
 

\[ r = r_{0} {\mathrm e}^{-t^{2}} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 14

ode=D[r[t],t]==-2*r[t]*t; 
ic={r[0]==r0}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to \text {r0} e^{-t^2} \end{align*}

✓ Sympy. Time used: 0.145 (sec). Leaf size: 8

from sympy import * 
t = symbols("t") 
r0 = symbols("r0") 
r = Function("r") 
ode = Eq(2*t*r(t) + Derivative(r(t), t),0) 
ics = {r(0): r0} 
dsolve(ode,func=r(t),ics=ics)
 

\[ r{\left (t \right )} = r_{0} e^{- t^{2}} \]