68.11.46 problem 58

Internal problem ID [17583]
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 : 58
Date solved : Thursday, October 02, 2025 at 02:25:44 PM
CAS classification : [[_2nd_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 23

ode:=diff(diff(y(t),t),t)-3*diff(y(t),t) = exp(-3*t)-exp(3*t); 
ic:=[y(0) = 1, D(y)(0) = 1]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \frac {4}{9}+\frac {\left (3-2 t \right ) {\mathrm e}^{3 t}}{6}+\frac {{\mathrm e}^{-3 t}}{18} \]

✓ Mathematica. Time used: 1.589 (sec). Leaf size: 73

ode=D[y[t],{t,2}]-3*D[y[t],t]==Exp[-3*t]-Exp[3*t]; 
ic={y[0]==1,Derivative[1][y][0] ==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \int _1^t\frac {1}{6} e^{-3 K[1]} \left (e^{6 K[1]} (7-6 K[1])-1\right )dK[1]-\int _1^0\frac {1}{6} e^{-3 K[1]} \left (e^{6 K[1]} (7-6 K[1])-1\right )dK[1]+1 \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(exp(3*t) - 3*Derivative(y(t), t) + Derivative(y(t), (t, 2)) - exp(-3*t),0) 
ics = {y(0): 1, Subs(Derivative(y(t), t), t, 0): 1} 
dsolve(ode,func=y(t),ics=ics)
 

NotImplementedError : The given ODE -2*sinh(3*t)/3 + Derivative(y(t), t) - Derivative(y(t), (t, 2))/3 cannot be solved by the factorable group method