12.19.44 problem section 9.3, problem 44

Internal problem ID [2191]
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 44
Date solved : Tuesday, March 04, 2025 at 01:51:25 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime \prime }+13 y^{\prime \prime }-19 y^{\prime }+10 y&={\mathrm e}^{x} \left (\left (7+8 x \right ) \cos \left (2 x \right )+\left (8-4 x \right ) \sin \left (2 x \right )\right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 50

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-5*diff(diff(diff(y(x),x),x),x)+13*diff(diff(y(x),x),x)-19*diff(y(x),x)+10*y(x) = exp(x)*((7+8*x)*cos(2*x)+(8-4*x)*sin(2*x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (x^{2}+x -4 c_4 +\frac {23}{4}\right ) {\mathrm e}^{x} \sin \left (2 x \right )}{4}+\frac {\left (4 c_3 +3\right ) {\mathrm e}^{x} \cos \left (2 x \right )}{4}+{\mathrm e}^{2 x} c_2 +\frac {{\mathrm e}^{x} \left (2 c_1 +7\right )}{2} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 172

ode=1*D[y[x],{x,4}]-5*D[y[x],{x,3}]+13*D[y[x],{x,2}]-19*D[y[x],x]-10*y[x]==Exp[x]*((7+8*x)*Cos[2*x]+(8-4*x)*Sin[2*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \exp \left (x \text {Root}\left [\text {$\#$1}^4-5 \text {$\#$1}^3+13 \text {$\#$1}^2-19 \text {$\#$1}-10\&,1\right ]\right )+c_3 \exp \left (x \text {Root}\left [\text {$\#$1}^4-5 \text {$\#$1}^3+13 \text {$\#$1}^2-19 \text {$\#$1}-10\&,3\right ]\right )+c_4 \exp \left (x \text {Root}\left [\text {$\#$1}^4-5 \text {$\#$1}^3+13 \text {$\#$1}^2-19 \text {$\#$1}-10\&,4\right ]\right )+c_2 \exp \left (x \text {Root}\left [\text {$\#$1}^4-5 \text {$\#$1}^3+13 \text {$\#$1}^2-19 \text {$\#$1}-10\&,2\right ]\right )-\frac {1}{100} e^x (-20 x \sin (2 x)+64 \sin (2 x)+40 x \cos (2 x)+67 \cos (2 x)) \]

✓ Sympy. Time used: 0.813 (sec). Leaf size: 34

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-(8 - 4*x)*sin(2*x) - (8*x + 7)*cos(2*x))*exp(x) + 10*y(x) - 19*Derivative(y(x), x) + 13*Derivative(y(x), (x, 2)) - 5*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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