26.8.15 problem Exercise 21.19, page 231

Internal problem ID [7099]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 4. Higher order linear differential equations. Lesson 21. Undetermined Coefficients
Problem number : Exercise 21.19, page 231
Date solved : Tuesday, September 30, 2025 at 04:21:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+3 y^{\prime }+2 y&={\mathrm e}^{-2 x}+x^{2} \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) = exp(-2*x)+x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {7}{4}-\left (1+x +c_1 \right ) {\mathrm e}^{-2 x}+\frac {x^{2}}{2}+{\mathrm e}^{-x} c_2 -\frac {3 x}{2} \]

✓ Mathematica. Time used: 0.124 (sec). Leaf size: 68

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

\begin{align*} y(x)&\to e^{-2 x} \left (\int _1^x\left (-e^{2 K[1]} K[1]^2-1\right )dK[1]+e^x \int _1^x\left (e^{K[2]} K[2]^2+e^{-K[2]}\right )dK[2]+c_2 e^x+c_1\right ) \end{align*}

✓ Sympy. Time used: 0.179 (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)) - exp(-2*x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} e^{- x} + \frac {x^{2}}{2} - \frac {3 x}{2} + \left (C_{1} - x\right ) e^{- 2 x} + \frac {7}{4} \]