76.2.30 problem 30

Internal problem ID [17295]
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 : 30
Date solved : Monday, March 31, 2025 at 03:49:48 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }-y&=1+3 \sin \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 20

ode:=diff(y(t),t)-y(t) = 1+3*sin(t); 
ic:=y(0) = y__0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -1-\frac {3 \cos \left (t \right )}{2}-\frac {3 \sin \left (t \right )}{2}+{\mathrm e}^{t} \left (y_{0} +\frac {5}{2}\right ) \]

✓ Mathematica. Time used: 0.074 (sec). Leaf size: 32

ode=D[y[t],t]-y[t]==1+3*Sin[t]; 
ic={y[0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^t \left (\int _0^te^{-K[1]} (3 \sin (K[1])+1)dK[1]+\text {y0}\right ) \]

✓ Sympy. Time used: 0.155 (sec). Leaf size: 26

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-y(t) - 3*sin(t) + Derivative(y(t), t) - 1,0) 
ics = {y(0): y__0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \left (y^{0} + \frac {5}{2}\right ) e^{t} - \frac {3 \sin {\left (t \right )}}{2} - \frac {3 \cos {\left (t \right )}}{2} - 1 \]