67.15.43 problem 22.11 (b)

Internal problem ID [16761]
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 (b)
Date solved : Thursday, October 02, 2025 at 01:38:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=x^{3} {\mathrm e}^{2 x} \sin \left (x \right ) \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 40

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

\[ y = -\frac {\left (\left (x^{4}-3 x^{2}-8 c_1 \right ) \cos \left (x \right )-2 \left (x^{3}-\frac {3}{2} x +4 c_2 \right ) \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{8} \]

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 64

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

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

✓ Sympy. Time used: 0.378 (sec). Leaf size: 39

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

\[ y{\left (x \right )} = \left (\left (C_{1} + \frac {x^{3}}{4} - \frac {3 x}{8}\right ) \sin {\left (x \right )} + \left (C_{2} - \frac {x^{4}}{8} + \frac {3 x^{2}}{8}\right ) \cos {\left (x \right )}\right ) e^{2 x} \]