86.11.13 problem 11 (b)

Internal problem ID [23207]
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 : 11 (b)
Date solved : Thursday, October 02, 2025 at 09:24:23 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }+4 y&={\mathrm e}^{x} \cos \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)+4*y(x) = exp(x)*cos(x); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {{\mathrm e}^{x} \left (\cos \left (\sqrt {3}\, x \right )-\cos \left (x \right )\right )}{2} \]

✓ Mathematica. Time used: 0.136 (sec). Leaf size: 25

ode=D[y[x],{x,2}]-2*D[y[x],x]+4*y[x]==Exp[x]*Cos[x]; 
ic={y[0]==0,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^x \left (\cos (x)-\cos \left (\sqrt {3} x\right )\right ) \end{align*}

✓ Sympy. Time used: 0.170 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = \left (\frac {\cos {\left (x \right )}}{2} - \frac {\cos {\left (\sqrt {3} x \right )}}{2}\right ) e^{x} \]