83.17.7 problem 7

Internal problem ID [19203]
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 : Tuesday, January 28, 2025 at 01:13:21 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 {\sqrt {3}\, x}{2}\right ) \end{align*}

✓ Solution by Maple

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

dsolve(diff(y(x),x$4)+diff(y(x),x$2)+y(x)=exp(-x/2)*cos(x*sqrt(3)/2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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