49.6.5 problem 1(e)

Internal problem ID [7633]
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 : Wednesday, March 05, 2025 at 04:48:52 AM
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: 0.004 (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: 0.288 (sec). Leaf size: 47

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]
 

\[ y(x)\to \frac {1}{250} \left (100 x^2+160 x+75 e^{-x}+88\right )+c_2 e^{2 x} \cos (x)+c_1 e^{2 x} \sin (x) \]

✓ Sympy. Time used: 0.301 (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)
 

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