83.17.7 problem 7

Internal problem ID [19124]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter III. Ordinary linear differential equations with constant coefficients. Misc. Examples on chapter III at page 50
Problem number : 7
Date solved : Thursday, March 13, 2025 at 01:44:29 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }+y&={\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {x \sqrt {3}}{2}\right ) \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 68

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+diff(diff(y(x),x),x)+y(x) = exp(-1/2*x)*cos(1/2*3^(1/2)*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y \left (x \right ) = \frac {\left (\left (6 x +24 c_{1} +11\right ) {\mathrm e}^{-\frac {x}{2}}+24 c_3 \,{\mathrm e}^{\frac {x}{2}}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{24}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \left (\left (\left (x -\frac {1}{2}\right ) \sqrt {3}+12 c_{2} \right ) {\mathrm e}^{-\frac {x}{2}}+12 c_4 \,{\mathrm e}^{\frac {x}{2}}\right )}{12} \]

✓ Mathematica. Time used: 1.069 (sec). Leaf size: 84

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

\[ y(x)\to \frac {1}{24} e^{-x/2} \left (\left (6 x+24 c_2 e^x+11+24 c_4\right ) \cos \left (\frac {\sqrt {3} x}{2}\right )+\left (2 \sqrt {3} x+24 c_3 e^x-\sqrt {3}+24 c_1\right ) \sin \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.398 (sec). Leaf size: 71

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

\[ y{\left (x \right )} = \left (C_{3} \sin {\left (\frac {\sqrt {3} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{\frac {x}{2}} + \left (\left (C_{1} + \frac {x}{4}\right ) \cos {\left (\frac {\sqrt {3} x}{2} \right )} + \left (C_{2} + \frac {\sqrt {3} x}{12}\right ) \sin {\left (\frac {\sqrt {3} x}{2} \right )}\right ) e^{- \frac {x}{2}} \]