76.2.28 problem 28

Internal problem ID [17293]
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 : 28
Date solved : Monday, March 31, 2025 at 03:49:43 PM
CAS classification : [[_linear, `class A`]]

\begin{align*} y^{\prime }+\frac {4 y}{3}&=1-\frac {t}{4} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 17

ode:=diff(y(t),t)+4/3*y(t) = 1-1/4*t; 
ic:=y(0) = y__0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = -\frac {3 t}{16}+\frac {57}{64}+{\mathrm e}^{-\frac {4 t}{3}} \left (y_{0} -\frac {57}{64}\right ) \]

✓ Mathematica. Time used: 0.084 (sec). Leaf size: 38

ode=D[y[t],t]+4/3*y[t]==1-1/4*t; 
ic={y[0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to e^{-4 t/3} \left (\int _0^t-\frac {1}{4} e^{\frac {4 K[1]}{3}} (K[1]-4)dK[1]+\text {y0}\right ) \]

✓ Sympy. Time used: 0.146 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t/4 + 4*y(t)/3 + Derivative(y(t), t) - 1,0) 
ics = {y(0): y__0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {3 t}{16} + \left (y^{0} - \frac {57}{64}\right ) e^{- \frac {4 t}{3}} + \frac {57}{64} \]