76.2.20 problem 20

Internal problem ID [17277]
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 : 20
Date solved : Thursday, March 13, 2025 at 09:22:55 AM
CAS classification : [_linear]

\begin{align*} t y^{\prime }+\left (1+t \right ) y&=t \end{align*}

With initial conditions

\begin{align*} y \left (\ln \left (2\right )\right )&=1 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 17

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

\[ y = \frac {2 \,{\mathrm e}^{-t}+t -1}{t} \]

✓ Mathematica. Time used: 0.063 (sec). Leaf size: 56

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

\[ y(t)\to \frac {e^{-t-1} \left (\int _1^te^{K[1]+1} K[1]dK[1]-\int _1^{\log (2)}e^{K[1]+1} K[1]dK[1]+e \log (4)\right )}{t} \]

✓ Sympy. Time used: 0.275 (sec). Leaf size: 14

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

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