1.11.18 problem 18

Internal problem ID [339]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 2. Linear Equations of Higher Order. Section 2.5 (Nonhomogeneous equations and undetermined coefficients). Problems at page 161
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 03:57:32 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime \prime }-y^{\prime \prime }+4 y&={\mathrm e}^{x}-x \,{\mathrm e}^{2 x} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 79

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-diff(diff(y(x),x),x)+4*y(x) = exp(x)-exp(2*x)*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left ({\mathrm e}^{-\frac {\sqrt {5}\, x}{2}} c_1 +{\mathrm e}^{\frac {\sqrt {5}\, x}{2}} c_3 \right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_2 \,{\mathrm e}^{-\frac {\sqrt {5}\, x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+c_4 \,{\mathrm e}^{\frac {\sqrt {5}\, x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )+\frac {\left (-4 x +7\right ) {\mathrm e}^{2 x}}{64}+\frac {{\mathrm e}^{x}}{4} \]

✓ Mathematica. Time used: 20.995 (sec). Leaf size: 18846

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

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✓ Sympy. Time used: 0.211 (sec). Leaf size: 143

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

\[ y{\left (x \right )} = - \frac {x e^{2 x}}{16} + \left (C_{1} \sin {\left (\sqrt {2} x \sin {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )} \right )} + C_{2} \cos {\left (\sqrt {2} x \sin {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )} \right )}\right ) e^{- \sqrt {2} x \cos {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )}} + \left (C_{3} \sin {\left (\sqrt {2} x \sin {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )} \right )} + C_{4} \cos {\left (\sqrt {2} x \sin {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )} \right )}\right ) e^{\sqrt {2} x \cos {\left (\frac {\operatorname {atan}{\left (\sqrt {15} \right )}}{2} \right )}} + \frac {7 e^{2 x}}{64} + \frac {e^{x}}{4} \]