70.2.22 problem 22

Internal problem ID [18645]
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 : 22
Date solved : Thursday, October 02, 2025 at 03:18:01 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 18

ode:=2*diff(y(t),t)-y(t) = exp(1/3*t); 
ic:=[y(0) = a]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \left (-3 \,{\mathrm e}^{-\frac {t}{6}}+a +3\right ) {\mathrm e}^{\frac {t}{2}} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 26

ode=2*D[y[t],t]-y[t]==Exp[t/3]; 
ic={y[0]==a}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{t/3} \left ((a+3) e^{t/6}-3\right ) \end{align*}

✓ Sympy. Time used: 0.095 (sec). Leaf size: 17

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

\[ y{\left (t \right )} = \left (a + 3\right ) e^{\frac {t}{2}} - 3 e^{\frac {t}{3}} \]