86.11.6 problem 6

Internal problem ID [23200]
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 : 6
Date solved : Thursday, October 02, 2025 at 09:24:17 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&=1+x^{2}+{\mathrm e}^{-2 x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

ode:=diff(diff(y(x),x),x)+3*diff(y(x),x)+2*y(x) = 1+x^2+exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {9}{4}-\left (1+x +c_1 \right ) {\mathrm e}^{-2 x}+\frac {x^{2}}{2}+{\mathrm e}^{-x} c_2 -\frac {3 x}{2} \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 41

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

\begin{align*} y(x)&\to \frac {1}{4} \left (2 x^2-6 x+9\right )+e^{-2 x} (-x-1+c_1)+c_2 e^{-x} \end{align*}

✓ Sympy. Time used: 0.198 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{2} e^{- x} + \frac {x^{2}}{2} - \frac {3 x}{2} + \left (C_{1} - x\right ) e^{- 2 x} + \frac {9}{4} \]