73.14.4 problem 21.6 (ii)

Internal problem ID [15335]
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 (ii)
Date solved : Thursday, March 13, 2025 at 05:55:09 AM
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 )&=1\\ y^{\prime }\left (0\right )&=-3 \end{align*}

✓ Maple. Time used: 0.170 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x)+2*diff(y(x),x)-8*y(x) = 8*x^2-3; 
ic:=y(0) = 1, D(y)(0) = -3; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.016 (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]==1,Derivative[1][y][0] ==-3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{4} e^{-4 x} \left (-2 e^{4 x} x (2 x+1)+e^{6 x}+3\right ) \]

✓ Sympy. Time used: 0.224 (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): 1, Subs(Derivative(y(x), x), x, 0): -3} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - x^{2} - \frac {x}{2} + \frac {e^{2 x}}{4} + \frac {3 e^{- 4 x}}{4} \]