68.11.38 problem 50

Internal problem ID [17575]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.3, page 156
Problem number : 50
Date solved : Thursday, October 02, 2025 at 02:25:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+y^{\prime }-6 y&=3 t \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {23}{12}} \\ y^{\prime }\left (0\right )&=-{\frac {3}{2}} \\ \end{align*}

✓ Maple. Time used: 0.037 (sec). Leaf size: 17

ode:=diff(diff(y(t),t),t)+diff(y(t),t)-6*y(t) = 3*t; 
ic:=[y(0) = 23/12, D(y)(0) = -3/2]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = {\mathrm e}^{2 t}+{\mathrm e}^{-3 t}-\frac {t}{2}-\frac {1}{12} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 24

ode=D[y[t],{t,2}]+D[y[t],t]-6*y[t]==3*t; 
ic={y[0]==23/12,Derivative[1][y][0] ==-3/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to -\frac {t}{2}+e^{-3 t}+e^{2 t}-\frac {1}{12} \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*t - 6*y(t) + Derivative(y(t), t) + Derivative(y(t), (t, 2)),0) 
ics = {y(0): 23/12, Subs(Derivative(y(t), t), t, 0): -3/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = - \frac {t}{2} + e^{2 t} - \frac {1}{12} + e^{- 3 t} \]