88.22.4 problem 8

Internal problem ID [24164]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Exercises at page 160 (Laplace transform)
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:00:22 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+16 y&=x^{2}-4 \cos \left (3 x \right ) \end{align*}

Using Laplace method

✓ Maple. Time used: 0.097 (sec). Leaf size: 102

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = x^2-4*cos(3*x); 
dsolve(ode,y(x),method='laplace');
 

\[ y = \frac {x^{2}}{16}-\frac {4 \cos \left (3 x \right )}{97}+\frac {\left (16 \cos \left (\sqrt {2}\, x \right ) \left (4+97 y \left (0\right )\right )+97 \sin \left (\sqrt {2}\, x \right ) \sqrt {2}\, \left (y^{\prime \prime \prime }\left (0\right )+4 y^{\prime }\left (0\right )\right )\right ) \cosh \left (\sqrt {2}\, x \right )}{1552}+\frac {\sqrt {2}\, \left (388 \left (-y^{\prime \prime \prime }\left (0\right )+4 y^{\prime }\left (0\right )\right ) \cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right ) \sqrt {2}\, \left (776 y^{\prime \prime }\left (0\right )-385\right )\right ) \sinh \left (\sqrt {2}\, x \right )}{6208} \]

✓ Mathematica. Time used: 0.805 (sec). Leaf size: 97

ode=D[y[x],{x,4}]+16*y[x]==x^2-4*Cos[3*x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {x^2}{16}-\frac {4}{97} \cos (3 x)+e^{-\sqrt {2} x} \left (c_1 e^{2 \sqrt {2} x}+c_2\right ) \cos \left (\sqrt {2} x\right )+c_3 e^{-\sqrt {2} x} \sin \left (\sqrt {2} x\right )+c_4 e^{\sqrt {2} x} \sin \left (\sqrt {2} x\right ) \end{align*}

✓ Sympy. Time used: 0.130 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x**2 + 16*y(x) + 4*cos(x) + Derivative(y(x), (x, 4)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2}}{16} + \left (C_{1} \sin {\left (\sqrt {2} x \right )} + C_{2} \cos {\left (\sqrt {2} x \right )}\right ) e^{- \sqrt {2} x} + \left (C_{3} \sin {\left (\sqrt {2} x \right )} + C_{4} \cos {\left (\sqrt {2} x \right )}\right ) e^{\sqrt {2} x} - \frac {4 \cos {\left (x \right )}}{17} \]