62.27.9 problem Ex 9

Internal problem ID [12947]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 50. Method of undetermined coefficients. Page 107
Problem number : Ex 9
Date solved : Tuesday, January 28, 2025 at 04:44:49 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 33

dsolve(diff(y(x),x$4)+2*diff(y(x),x$2)+y(x)=cos(x),y(x), singsol=all)
 

\[ y = \frac {\left (8 c_4 x -x^{2}+8 c_{1} +2\right ) \cos \left (x \right )}{8}+\sin \left (x \right ) \left (\left (c_3 +\frac {1}{8}\right ) x +c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.136 (sec). Leaf size: 115

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

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