23.5.14 problem 14

Internal problem ID [6623]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 14
Date solved : Tuesday, September 30, 2025 at 03:50:19 PM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 35

ode:=4*y(x)-2*diff(y(x),x)+diff(diff(diff(y(x),x),x),x) = exp(x)*cos(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,{\mathrm e}^{-2 x}-\frac {\left (\left (x -20 c_2 -\frac {18}{5}\right ) \cos \left (x \right )-3 \sin \left (x \right ) \left (x +\frac {20 c_3}{3}+\frac {1}{15}\right )\right ) {\mathrm e}^{x}}{20} \]

✓ Mathematica. Time used: 0.033 (sec). Leaf size: 53

ode=4*y[x] - 2*D[y[x],x] + D[y[x],{x,3}] == E^x*Cos[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_3 e^{-2 x}-\frac {1}{100} e^x (5 x-2 (9+50 c_2)) \cos (x)+\frac {1}{100} e^x (15 x+1+100 c_1) \sin (x) \end{align*}

✓ Sympy. Time used: 0.199 (sec). Leaf size: 31

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

\[ y{\left (x \right )} = C_{3} e^{- 2 x} + \left (\left (C_{1} - \frac {x}{20}\right ) \cos {\left (x \right )} + \left (C_{2} + \frac {3 x}{20}\right ) \sin {\left (x \right )}\right ) e^{x} \]