76.2.19 problem 19

Internal problem ID [17276]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.2 (Linear equations: Method of integrating factors). Problems at page 54
Problem number : 19
Date solved : Thursday, March 13, 2025 at 09:22:53 AM
CAS classification : [_linear]

\begin{align*} t^{3} y^{\prime }+4 t^{2} y&={\mathrm e}^{-t} \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 16

ode:=t^3*diff(y(t),t)+4*t^2*y(t) = exp(-t); 
ic:=y(-1) = 0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {\left (t +1\right ) {\mathrm e}^{-t}}{t^{4}} \]

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 25

ode=t^3*D[y[t],t]+4*t^2*y[t]==Exp[-t]; 
ic={y[-1]==0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {\int _{-1}^te^{-K[1]} K[1]dK[1]}{t^4} \]

✓ Sympy. Time used: 0.281 (sec). Leaf size: 19

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t**3*Derivative(y(t), t) + 4*t**2*y(t) - exp(-t),0) 
ics = {y(-1): 0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {- e^{- t} - \frac {e^{- t}}{t}}{t^{3}} \]