63.1.113 problem 160

Internal problem ID [15553]
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 : Thursday, October 02, 2025 at 10:19:49 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 52

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

\[ y = \frac {\left (2+\left (c_3 x +c_1 \right ) a^{4}-a^{2} x^{2}\right ) \cos \left (a x \right )+\sin \left (a x \right ) a \left (\left (c_4 x +c_2 \right ) a^{3}+3 x \right )}{a^{4}} \]

✓ Mathematica. Time used: 0.195 (sec). Leaf size: 161

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

\begin{align*} 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) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**4*y(x) + 2*a**2*Derivative(y(x), (x, 2)) - 8*cos(a*x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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