85.56.2 problem 1 (b)

Internal problem ID [22827]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 199
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 09:15:03 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 62

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

\[ y = \frac {\left (32 x^{4}-32 x^{3}+24 x^{2}+512 c_2 -12 x +3\right ) {\mathrm e}^{3 x}}{512}+\frac {4 \left (65 x -61\right ) \cos \left (2 x \right )}{4225}+\frac {\left (-455 x -158\right ) \sin \left (2 x \right )}{4225}+{\mathrm e}^{-x} c_1 \]

✓ Mathematica. Time used: 0.211 (sec). Leaf size: 73

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

\begin{align*} y(x)&\to \frac {1}{512} e^{3 x} \left (32 x^4-32 x^3+24 x^2-12 x+3+512 c_2\right )-\frac {(455 x+158) \sin (2 x)}{4225}+\frac {4 (65 x-61) \cos (2 x)}{4225}+c_1 e^{-x} \end{align*}

✓ Sympy. Time used: 0.326 (sec). Leaf size: 107

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

\[ y{\left (x \right )} = C_{1} e^{\frac {x \left (1 - \sqrt {13}\right )}{2}} + C_{2} e^{\frac {x \left (1 + \sqrt {13}\right )}{2}} + \frac {x^{3} e^{3 x}}{3} - \frac {5 x^{2} e^{3 x}}{3} + \frac {44 x e^{3 x}}{9} - \frac {7 x \sin {\left (2 x \right )}}{53} + \frac {2 x \cos {\left (2 x \right )}}{53} - \frac {190 e^{3 x}}{27} - \frac {67 \sin {\left (2 x \right )}}{2809} - \frac {208 \cos {\left (2 x \right )}}{2809} \]