12.19.67 problem section 9.3, problem 67

Internal problem ID [2214]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number : section 9.3, problem 67
Date solved : Tuesday, March 04, 2025 at 01:51:50 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }-3 y^{\prime \prime \prime }+4 y^{\prime \prime }-2 y^{\prime }&={\mathrm e}^{x} \left (\left (28+6 x \right ) \cos \left (2 x \right )+\left (11-12 x \right ) \sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 34

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-3*diff(diff(diff(y(x),x),x),x)+4*diff(diff(y(x),x),x)-2*diff(y(x),x) = exp(x)*((28+6*x)*cos(2*x)+(11-12*x)*sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (-4 x \sin \left (x \right )+c_2 -c_3 \right ) \cos \left (x \right )+\left (c_2 +c_3 \right ) \sin \left (x \right )+2 c_1 +3\right ) {\mathrm e}^{x}}{2}+c_4 \]

✓ Mathematica. Time used: 0.339 (sec). Leaf size: 43

ode=D[y[x],{x,4}]-3*D[y[x],{x,3}]+4*D[y[x],{x,2}]-2*D[y[x],x]-0*y[x]==Exp[x]*((28+6*x)*Cos[2*x]+(11-12*x)*Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^x ((c_1+c_2) \sin (x)+\cos (x) (-4 x \sin (x)-c_1+c_2)+2 c_3)+c_4 \]

✓ Sympy. Time used: 0.573 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-(11 - 12*x)*sin(2*x) - (6*x + 28)*cos(2*x))*exp(x) - 2*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)) - 3*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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