67.8.38 problem 13.6 (d)

Internal problem ID [16533]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 13. Higher order equations: Extending first order concepts. Additional exercises page 259
Problem number : 13.6 (d)
Date solved : Thursday, October 02, 2025 at 01:36:08 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }&=8 \,{\mathrm e}^{2 x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 10

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x) = 8*exp(2*x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 4 \sinh \left (x \right )^{2} \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 17

ode=D[y[x],{x,2}]+2*D[y[x],x]==8*Exp[2*x]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{-2 x}+e^{2 x}-2 \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*exp(2*x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{2 x} - 2 + e^{- 2 x} \]