67.14.3 problem 21.6 (i)

Internal problem ID [16700]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 21. Nonhomogeneous equations in general. Additional exercises page 391
Problem number : 21.6 (i)
Date solved : Thursday, October 02, 2025 at 01:37:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 y^{\prime }-8 y&=8 x^{2}-3 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)-8*y(x) = 8*x^2-3; 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{-4 x}}{12}+\frac {{\mathrm e}^{2 x}}{12}-x^{2}-\frac {x}{2} \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 34

ode=D[y[x],{x,2}]+2*D[y[x],x]-8*y[x]==8*x^2-3; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{12} e^{-4 x} \left (-6 e^{4 x} x (2 x+1)+e^{6 x}-1\right ) \end{align*}

✓ Sympy. Time used: 0.131 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**2 - 8*y(x) + 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 3,0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x^{2} - \frac {x}{2} + \frac {e^{2 x}}{12} - \frac {e^{- 4 x}}{12} \]