43.11.3 problem 1(c)

Internal problem ID [8953]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 2. Linear equations with constant coefficients. Page 93
Problem number : 1(c)
Date solved : Tuesday, September 30, 2025 at 06:00:29 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

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

\[ y = {\mathrm e}^{-2 x} \left (\frac {{\mathrm e}^{4 x} \left (-3+16 c_1 +12 x \right )}{16}+c_2 -\frac {4 \,{\mathrm e}^{x}}{3}\right ) \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 86

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

\begin{align*} y(x)&\to e^{-2 x} \left (e^{4 x} \int _1^x\frac {1}{4} e^{-3 K[1]} \left (3 e^{K[1]} \exp (2 K[1])+4\right )dK[1]+\int _1^x-\frac {1}{4} e^{K[2]} \left (3 e^{K[2]} \exp (2 K[2])+4\right )dK[2]+c_1 e^{4 x}+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.074 (sec). Leaf size: 27

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

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