6.19.47 problem section 9.3, problem 47

Internal problem ID [2194]
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 47
Date solved : Tuesday, September 30, 2025 at 05:24:51 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+26 y^{\prime \prime }-40 y^{\prime }+25 y&={\mathrm e}^{2 x} \left (3 \cos \left (x \right )-\left (1+3 x \right ) \sin \left (x \right )\right ) \end{align*}

✓ Maple. Time used: 0.083 (sec). Leaf size: 44

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-8*diff(diff(diff(y(x),x),x),x)+26*diff(diff(y(x),x),x)-40*diff(y(x),x)+25*y(x) = exp(2*x)*(3*cos(x)-(1+3*x)*sin(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {{\mathrm e}^{2 x} \left (\left (x^{3}+x^{2}+\left (8 c_4 +3\right ) x +8 c_2 -2\right ) \sin \left (x \right )+8 \left (\left (c_3 +\frac {1}{4}\right ) x +c_1 +\frac {9}{16}\right ) \cos \left (x \right )\right )}{8} \]

✓ Mathematica. Time used: 0.143 (sec). Leaf size: 60

ode=1*D[y[x],{x,4}]-8*D[y[x],{x,3}]+26*D[y[x],{x,2}]-40*D[y[x],x]+25*y[x]==Exp[2*x]*(3*Cos[1*x]-(1+3*x)*Sin[1*x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{16} e^{2 x} \left (\left (2 x^3+2 x^2+(9+16 c_2) x-1+16 c_1\right ) \sin (x)+(2 (1+8 c_4) x+3+16 c_3) \cos (x)\right ) \end{align*}

✓ Sympy. Time used: 0.602 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(((3*x + 1)*sin(x) - 3*cos(x))*exp(2*x) + 25*y(x) - 40*Derivative(y(x), x) + 26*Derivative(y(x), (x, 2)) - 8*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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