85.33.54 problem 54

Internal problem ID [22677]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 54
Date solved : Thursday, October 02, 2025 at 09:06:15 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} u^{\prime }&=-a \left (u-100 t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 23

ode:=diff(u(t),t) = -a*(u(t)-100*t); 
ic:=[u(0) = 0]; 
dsolve([ode,op(ic)],u(t), singsol=all);
 

\[ u = 100 t -\frac {100}{a}+\frac {100 \,{\mathrm e}^{-a t}}{a} \]

✓ Mathematica. Time used: 0.034 (sec). Leaf size: 29

ode=D[u[t],t]== -a*(u[t]-100*t); 
ic={u[0]==0}; 
DSolve[{ode,ic},u[t],t,IncludeSingularSolutions->True]
 

\begin{align*} u(t)&\to \frac {100 e^{-a t} \left (e^{a t} (a t-1)+1\right )}{a} \end{align*}

✓ Sympy. Time used: 0.090 (sec). Leaf size: 17

from sympy import * 
t = symbols("t") 
a = symbols("a") 
u = Function("u") 
ode = Eq(a*(-100*t + u(t)) + Derivative(u(t), t),0) 
ics = {u(0): 0} 
dsolve(ode,func=u(t),ics=ics)
 

\[ u{\left (t \right )} = 100 t - \frac {100}{a} + \frac {100 e^{- a t}}{a} \]