67.15.52 problem 22.11 (k)

Internal problem ID [16770]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number : 22.11 (k)
Date solved : Thursday, October 02, 2025 at 01:38:41 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+20 y&={\mathrm e}^{2 x} \sin \left (4 x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+20*y(x) = exp(2*x)*sin(4*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\left (x -8 c_1 \right ) \cos \left (4 x \right )-8 c_2 \sin \left (4 x \right )\right ) {\mathrm e}^{2 x}}{8} \]

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 70

ode=D[y[x],{x,2}]-4*D[y[x],x]+20*y[x]==Exp[2*x]*Sin[4*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{2 x} \left (\sin (4 x) \int _1^x\frac {1}{8} \sin (8 K[1])dK[1]+\cos (4 x) \int _1^x-\frac {1}{4} \sin ^2(4 K[2])dK[2]+c_2 \cos (4 x)+c_1 \sin (4 x)\right ) \end{align*}

✓ Sympy. Time used: 0.243 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(20*y(x) - exp(2*x)*sin(4*x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{2} \sin {\left (4 x \right )} + \left (C_{1} - \frac {x}{8}\right ) \cos {\left (4 x \right )}\right ) e^{2 x} \]