76.2.23 problem 23

Internal problem ID [17288]
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 : 23
Date solved : Monday, March 31, 2025 at 03:48:54 PM
CAS classification : [[_linear, `class A`]]

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

With initial conditions

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 34

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

\[ y = \frac {{\mathrm e}^{\frac {2 t}{3}} \left (3 \pi a +4 a +2\right )-2 \,{\mathrm e}^{-\frac {\pi t}{2}}}{3 \pi +4} \]

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 43

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

\[ y(t)\to \frac {e^{2 t/3} \left ((4+3 \pi ) a-2 e^{-\frac {1}{6} (4+3 \pi ) t}+2\right )}{4+3 \pi } \]

✓ Sympy. Time used: 0.237 (sec). Leaf size: 37

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

\[ y{\left (t \right )} = \frac {\left (4 a + 3 \pi a + 2\right ) e^{\frac {2 t}{3}}}{4 + 3 \pi } - \frac {2 e^{- \frac {\pi t}{2}}}{4 + 3 \pi } \]