64.11.24 problem 24

Internal problem ID [13395]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 24
Date solved : Wednesday, March 05, 2025 at 09:51:56 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-5 y^{\prime \prime \prime }+7 y^{\prime \prime }-5 y^{\prime }+6 y&=5 \sin \left (x \right )-12 \sin \left (2 x \right ) \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 51

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-5*diff(diff(diff(y(x),x),x),x)+7*diff(diff(y(x),x),x)-5*diff(y(x),x)+6*y(x) = 5*sin(x)-12*sin(2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {5 \cos \left (2 x \right )}{13}+c_{3} {\mathrm e}^{2 x}+c_4 \,{\mathrm e}^{3 x}+\frac {\sin \left (2 x \right )}{13}+\frac {\left (-2-5 x +20 c_{1} \right ) \cos \left (x \right )}{20}+\frac {\left (1+x +4 c_{2} \right ) \sin \left (x \right )}{4} \]

✓ Mathematica. Time used: 0.434 (sec). Leaf size: 168

ode=D[y[x],{x,4}]-5*D[y[x],{x,3}]+7*D[y[x],{x,2}]-5*D[y[x],x]+6*y[x]==5*Sin[x]-12*Sin[2*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \cos (x) \int _1^x\frac {1}{10} (\cos (K[1])-\sin (K[1])) (5 \sin (K[1])-12 \sin (2 K[1]))dK[1]+\sin (x) \int _1^x\frac {1}{10} (\cos (K[2])+\sin (K[2])) (5 \sin (K[2])-12 \sin (2 K[2]))dK[2]+e^{2 x} \int _1^x\frac {1}{5} e^{-2 K[3]} (24 \cos (K[3])-5) \sin (K[3])dK[3]+e^{3 x} \int _1^x\frac {1}{10} e^{-3 K[4]} (5-24 \cos (K[4])) \sin (K[4])dK[4]+c_3 e^{2 x}+c_4 e^{3 x}+c_1 \cos (x)+c_2 \sin (x) \]

✓ Sympy. Time used: 0.390 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(6*y(x) - 5*sin(x) + 12*sin(2*x) - 5*Derivative(y(x), x) + 7*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)
 

\[ y{\left (x \right )} = C_{3} e^{2 x} + C_{4} e^{3 x} + \left (C_{1} - \frac {x}{4}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {x}{4}\right ) \sin {\left (x \right )} + \frac {\sin {\left (2 x \right )}}{13} + \frac {5 \cos {\left (2 x \right )}}{13} \]