62.29.2 problem Ex 2

Internal problem ID [12882]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number : Ex 2
Date solved : Monday, March 31, 2025 at 07:23:32 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y&={\mathrm e}^{x} \cos \left (x \right ) \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 31

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-y(x) = exp(x)*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_4 \,{\mathrm e}^{-x}+\frac {\left (5 c_1 -{\mathrm e}^{x}\right ) \cos \left (x \right )}{5}+c_2 \,{\mathrm e}^{x}+c_3 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.057 (sec). Leaf size: 120

ode=D[y[x],{x,4}]-y[x]==Exp[x]*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.144 (sec). Leaf size: 27

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

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