68.5.46 problem 53

Internal problem ID [17297]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.3, page 49
Problem number : 53
Date solved : Thursday, October 02, 2025 at 02:01:06 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} -\frac {y}{2}+y^{\prime }&=5 \cos \left (t \right )+2 \,{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(y(t),t)-1/2*y(t) = 5*cos(t)+2*exp(t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = 4 \sin \left (t \right )-2 \cos \left (t \right )+4 \,{\mathrm e}^{t}+{\mathrm e}^{\frac {t}{2}} c_1 \]

✓ Mathematica. Time used: 0.102 (sec). Leaf size: 44

ode=D[y[t],t]-1/2*y[t]==5*Cos[t]+2*Exp[t]; 
ic={}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to e^{t/2} \left (\int _1^te^{-\frac {K[1]}{2}} \left (5 \cos (K[1])+2 e^{K[1]}\right )dK[1]+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t)/2 - 2*exp(t) - 5*cos(t) + Derivative(y(t), t),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} e^{\frac {t}{2}} + 4 e^{t} + 4 \sin {\left (t \right )} - 2 \cos {\left (t \right )} \]