23.5.161 problem 161

Internal problem ID [6770]
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 : 161
Date solved : Tuesday, September 30, 2025 at 03:51:31 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 133

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

\[ y = \frac {\left (-5 a^{2}+b^{2}\right ) \cos \left (a x \right )}{8 \left (a +b \right )^{2} \left (a -b \right )^{2} a^{2}}+\frac {\left (a^{2}-5 b^{2}\right ) \cos \left (b x \right )}{8 \left (a +b \right )^{2} \left (a -b \right )^{2} b^{2}}-\frac {x \sin \left (a x \right )}{4 a \left (a -b \right ) \left (a +b \right )}+\frac {x \sin \left (b x \right )}{4 b \left (a -b \right ) \left (a +b \right )}+c_1 \cos \left (a x \right )+c_2 \cos \left (b x \right )+c_3 \sin \left (a x \right )+c_4 \sin \left (b x \right ) \]

✓ Mathematica. Time used: 0.288 (sec). Leaf size: 173

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

\begin{align*} y(x)&\to \frac {a \left (2 b \left (b^2-a^2\right ) \left (b \sin (a x) \left (x+4 a c_4 \left (b^2-a^2\right )\right )-a \sin (b x) \left (x+4 b c_2 \left (a^2-b^2\right )\right )\right )+a \left (8 a^4 b^2 c_1-16 a^2 b^4 c_1+a^2+8 b^6 c_1-5 b^2\right ) \cos (b x)\right )+b^2 \left (8 a^6 c_3-16 a^4 b^2 c_3+a^2 \left (-5+8 b^4 c_3\right )+b^2\right ) \cos (a x)}{8 a^2 b^2 (a-b)^2 (a+b)^2} \end{align*}

✓ Sympy. Time used: 0.233 (sec). Leaf size: 90

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

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