76.2.24 problem 24

Internal problem ID [17360]
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 : Tuesday, January 28, 2025 at 10:02:22 AM
CAS classification : [_linear]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 21

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

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

✓ Solution by Mathematica

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

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

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