60.6.2 problem 1579

Internal problem ID [11539]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 5, linear fifth and higher order
Problem number : 1579
Date solved : Sunday, March 30, 2025 at 08:24:27 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\left (5\right )}+2 y^{\prime \prime \prime }+y^{\prime }-a x -b \sin \left (x \right )-c \cos \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 69

ode:=diff(diff(diff(diff(diff(y(x),x),x),x),x),x)+2*diff(diff(diff(y(x),x),x),x)+diff(y(x),x)-a*x-b*sin(x)-c*cos(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (b \,x^{2}+\left (-4 c -8 c_4 \right ) x -6 b -8 c_2 +8 c_3 \right ) \cos \left (x \right )}{8}+\frac {\left (-c \,x^{2}+\left (-4 b +8 c_3 \right ) x +6 c +8 c_1 +8 c_4 \right ) \sin \left (x \right )}{8}+\frac {x^{2} a}{2}+c_5 \]

✓ Mathematica. Time used: 0.596 (sec). Leaf size: 200

ode=D[y[x],{x,5}]+2*D[y[x],{x,3}]+D[y[x],x]-a*x-b*Sin[x]-c*Cos[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\left (c_1 \cos (K[5])+c_2 K[5] \cos (K[5])+\int _1^{K[5]}\frac {1}{2} (\cos (K[1]) K[1]-\sin (K[1])) (c \cos (K[1])+a K[1]+b \sin (K[1]))dK[1] \cos (K[5])+K[5] \int _1^{K[5]}-\frac {1}{2} \cos (K[2]) (c \cos (K[2])+a K[2]+b \sin (K[2]))dK[2] \cos (K[5])+c_3 \sin (K[5])+c_4 K[5] \sin (K[5])+\sin (K[5]) \int _1^{K[5]}\frac {1}{2} (c \cos (K[3])+a K[3]+b \sin (K[3])) (\cos (K[3])+K[3] \sin (K[3]))dK[3]+K[5] \sin (K[5]) \int _1^{K[5]}-\frac {1}{2} \sin (K[4]) (c \cos (K[4])+a K[4]+b \sin (K[4]))dK[4]\right )dK[5]+c_5 \]

✓ Sympy. Time used: 0.327 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-a*x - b*sin(x) - c*cos(x) + Derivative(y(x), x) + 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 5)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + \frac {a x^{2}}{2} + \left (C_{2} + x \left (C_{3} - \frac {c x}{8}\right )\right ) \sin {\left (x \right )} + \left (C_{4} + x \left (C_{5} + \frac {b x}{8}\right )\right ) \cos {\left (x \right )} \]