69.1.113 problem 160

Internal problem ID [14266]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 160
Date solved : Tuesday, January 28, 2025 at 06:24:56 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+2 a^{2} y^{\prime \prime }+a^{4} y&=8 \cos \left (a x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 52

dsolve(diff(y(x),x$4)+2*a^2*diff(y(x),x$2)+a^4*y(x)=8*cos(a*x),y(x), singsol=all)
 

\[ y = \frac {\left (2+\left (c_{3} x +c_{1} \right ) a^{4}-a^{2} x^{2}\right ) \cos \left (a x \right )+a \left (\left (c_4 x +c_{2} \right ) a^{3}+3 x \right ) \sin \left (a x \right )}{a^{4}} \]

✓ Solution by Mathematica

Time used: 0.327 (sec). Leaf size: 161

DSolve[D[y[x],{x,4}]+2*a^2*D[y[x],{x,2}]+a^4*y[x]==8*Cos[a*x],y[x],x,IncludeSingularSolutions -> True]
 

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