86.11.12 problem 11 (a)

Internal problem ID [23206]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 9. The operational method. Exercise 9c at page 137
Problem number : 11 (a)
Date solved : Thursday, October 02, 2025 at 09:24:22 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 16 y+8 y^{\prime }+y^{\prime \prime }&=x \left (12-{\mathrm e}^{-4 x}\right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)+8*diff(y(x),x)+16*y(x) = x*(12-exp(-4*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {3}{8}+\frac {\left (-x^{3}+6 c_1 x +6 c_2 \right ) {\mathrm e}^{-4 x}}{6}+\frac {3 x}{4} \]

✓ Mathematica. Time used: 0.147 (sec). Leaf size: 37

ode=D[y[x],{x,2}]+8*D[y[x],x]+16*y[x]==x*(12-Exp[-4*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {3}{8} (2 x-1)-\frac {1}{6} e^{-4 x} \left (x^3-6 c_2 x-6 c_1\right ) \end{align*}

✓ Sympy. Time used: 0.204 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(12 - exp(-4*x)) + 16*y(x) + 8*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {3 x}{4} + \left (C_{1} + x \left (C_{2} - \frac {x^{2}}{6}\right )\right ) e^{- 4 x} - \frac {3}{8} \]