28.2.19 problem 19

Internal problem ID [4462]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 4. Linear Differential Equations. Page 183
Problem number : 19
Date solved : Sunday, March 30, 2025 at 03:23:14 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-2 y^{\prime \prime }+4 y^{\prime }-8 y&={\mathrm e}^{2 x} \sin \left (2 x \right )+2 x^{2} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 58

ode:=diff(diff(diff(y(x),x),x),x)-2*diff(diff(y(x),x),x)+4*diff(y(x),x)-8*y(x) = exp(2*x)*sin(2*x)+2*x^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (80 c_2 -2 \cos \left (2 x \right )-4 \sin \left (2 x \right )-5\right ) {\mathrm e}^{2 x}}{80}+\frac {\left (80 c_1 -5\right ) \cos \left (2 x \right )}{80}+\frac {\left (80 c_3 +5\right ) \sin \left (2 x \right )}{80}-\frac {x^{2}}{4}-\frac {x}{4} \]

✓ Mathematica. Time used: 0.249 (sec). Leaf size: 61

ode=D[y[x],{x,3}]-2*D[y[x],{x,2}]+4*D[y[x],x]-8*y[x]==Exp[2*x]*Sin[2*x]+2*x^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{80} \left (-20 x (x+1)+5 (-1+16 c_3) e^{2 x}-2 \left (e^{2 x}-40 c_1\right ) \cos (2 x)-4 \left (e^{2 x}-20 c_2\right ) \sin (2 x)\right ) \]

✓ Sympy. Time used: 0.367 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x**2 - 8*y(x) - exp(2*x)*sin(2*x) + 4*Derivative(y(x), x) - 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{2} \sin {\left (2 x \right )} + C_{3} \cos {\left (2 x \right )} - \frac {x^{2}}{4} - \frac {x}{4} + \left (C_{1} - \frac {\sin {\left (2 x \right )}}{20} - \frac {\cos {\left (2 x \right )}}{40}\right ) e^{2 x} \]