43.6.5 problem 1(e)

Internal problem ID [8919]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 69
Problem number : 1(e)
Date solved : Tuesday, September 30, 2025 at 06:00:09 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=3 \,{\mathrm e}^{-x}+2 x^{2} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+5*y(x) = 3*exp(-x)+2*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x} \sin \left (x \right ) c_2 +{\mathrm e}^{2 x} \cos \left (x \right ) c_1 +\frac {3 \,{\mathrm e}^{-x}}{10}+\frac {2 x^{2}}{5}+\frac {16 x}{25}+\frac {44}{125} \]

✓ Mathematica. Time used: 0.108 (sec). Leaf size: 87

ode=D[y[x],{x,2}]-4*D[y[x],x]+5*y[x]==3*Exp[-x]+2*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{2 x} \left (\cos (x) \int _1^x-e^{-3 K[2]} \left (2 e^{K[2]} K[2]^2+3\right ) \sin (K[2])dK[2]+\sin (x) \int _1^xe^{-3 K[1]} \cos (K[1]) \left (2 e^{K[1]} K[1]^2+3\right )dK[1]+c_2 \cos (x)+c_1 \sin (x)\right ) \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 + 5*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 3*exp(-x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {2 x^{2}}{5} + \frac {16 x}{25} + \left (C_{1} \sin {\left (x \right )} + C_{2} \cos {\left (x \right )}\right ) e^{2 x} + \frac {44}{125} + \frac {3 e^{- x}}{10} \]