36.2.4 problem 4

Internal problem ID [6297]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 2, First order differential equations. Section 2.3, Linear equations. Exercises. page 54
Problem number : 4
Date solved : Sunday, March 30, 2025 at 10:49:50 AM
CAS classification : [_linear]

\begin{align*} 3 t&={\mathrm e}^{t} y^{\prime }+y \ln \left (t \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 42

ode:=3*t = exp(t)*diff(y(t),t)+y(t)*ln(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \left (3 \int t^{1-{\mathrm e}^{-t}} {\mathrm e}^{-t -\operatorname {Ei}_{1}\left (t \right )}d t +c_1 \right ) t^{{\mathrm e}^{-t}} {\mathrm e}^{\operatorname {Ei}_{1}\left (t \right )} \]

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 58

ode=3*t==Exp[t]*D[y[t],t]+y[t]*Log[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to t^{e^{-t}} e^{-\operatorname {ExpIntegralEi}(-t)} \left (\int _1^t3 e^{\operatorname {ExpIntegralEi}(-K[1])-K[1]} K[1]^{1-e^{-K[1]}}dK[1]+c_1\right ) \]

✓ Sympy. Time used: 28.345 (sec). Leaf size: 51

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

\[ - 3 \int t e^{- t} e^{- e^{- t} \log {\left (t \right )}} e^{\operatorname {Ei}{\left (- t \right )}}\, dt + \int y{\left (t \right )} e^{- t} e^{- e^{- t} \log {\left (t \right )}} e^{\operatorname {Ei}{\left (- t \right )}} \log {\left (t \right )}\, dt = C_{1} \]