23.5.185 problem 185

Internal problem ID [6794]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 185
Date solved : Tuesday, September 30, 2025 at 03:51:44 PM
CAS classification : [[_high_order, _missing_y]]

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

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

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

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

✓ Mathematica. Time used: 0.428 (sec). Leaf size: 80

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

\begin{align*} y(x)&\to \frac {1}{16} \left (8 a x^2+\cos (x) \left (c \left (2 x^2-9\right )-2 (5 b x+8 (c_4 x-c_2+c_3))\right )+\sin (x) \left (b \left (13-2 x^2\right )-6 c x+16 (c_2 x+c_1+c_4)\right )\right )+c_5 \end{align*}

✓ Sympy. Time used: 0.199 (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*cos(x) - c*sin(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)
 

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