76.2.24 problem 24

Internal problem ID [17289]
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 : 24
Date solved : Monday, March 31, 2025 at 03:48:56 PM
CAS classification : [_linear]

\begin{align*} t y^{\prime }+\left (1+t \right ) y&=2 t \,{\mathrm e}^{-t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 21

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

\[ y = \frac {\left (t^{2}-1+{\mathrm e} a \right ) {\mathrm e}^{-t}}{t} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 22

ode=t*D[y[t],t]+(1+t)*y[t]==2*t*Exp[-t]; 
ic={y[1]==a}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {e^{-t} \left (e a+t^2-1\right )}{t} \]

✓ Sympy. Time used: 0.291 (sec). Leaf size: 15

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

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