58.11.52 problem 52

Internal problem ID [14783]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number : 52
Date solved : Thursday, October 02, 2025 at 09:54:48 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }+16 y&=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \end{align*}

✓ Maple. Time used: 0.023 (sec). Leaf size: 108

ode:=diff(diff(diff(diff(y(x),x),x),x),x)+16*y(x) = x*exp(2^(1/2)*x)*sin(2^(1/2)*x)+exp(-2^(1/2)*x)*cos(2^(1/2)*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\left (2 \sqrt {2}\, x +128 c_3 +3\right ) \cos \left (\sqrt {2}\, x \right )+2 \sin \left (\sqrt {2}\, x \right ) \left (\sqrt {2}\, x +64 c_4 \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{128}-\frac {\left (\left (x^{2} \sqrt {2}-128 c_1 -\frac {5 \sqrt {2}}{8}\right ) \cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right ) \left (x^{2} \sqrt {2}-3 x -128 c_2 +\frac {5 \sqrt {2}}{8}\right )\right ) {\mathrm e}^{\sqrt {2}\, x}}{128} \]

✓ Mathematica. Time used: 1.586 (sec). Leaf size: 140

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

\begin{align*} y(x)&\to \frac {e^{-\sqrt {2} x} \left (\left (e^{2 \sqrt {2} x} \left (-8 \sqrt {2} x^2+5 \sqrt {2}+1024 c_1\right )+8 \left (2 \sqrt {2} x+3+128 c_2\right )\right ) \cos \left (\sqrt {2} x\right )-\left (e^{2 \sqrt {2} x} \left (8 \sqrt {2} x^2-24 x+5 \sqrt {2}-1024 c_4\right )-16 \left (\sqrt {2} x+64 c_3\right )\right ) \sin \left (\sqrt {2} x\right )\right )}{1024} \end{align*}

✓ Sympy. Time used: 0.397 (sec). Leaf size: 102

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

\[ y{\left (x \right )} = \left (\left (C_{1} - \frac {\sqrt {2} x^{2}}{128}\right ) \cos {\left (\sqrt {2} x \right )} + \left (C_{2} - \frac {\sqrt {2} x^{2}}{128} + \frac {3 x}{128}\right ) \sin {\left (\sqrt {2} x \right )}\right ) e^{\sqrt {2} x} + \left (\left (C_{3} + \frac {\sqrt {2} x}{64}\right ) \sin {\left (\sqrt {2} x \right )} + \left (C_{4} + \frac {\sqrt {2} x}{64}\right ) \cos {\left (\sqrt {2} x \right )}\right ) e^{- \sqrt {2} x} \]