82.31.1 problem Ex. 1

Internal problem ID [18818]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VI. Linear equations with constant coefficients. problems at page 79
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:15:40 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 78

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

\[ y = \frac {\left (289 c_4 \,{\mathrm e}^{\frac {\sqrt {2}\, x}{2}}+289 c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, x}{2}}\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{289}+c_3 \,{\mathrm e}^{\frac {\sqrt {2}\, x}{2}} \cos \left (\frac {\sqrt {2}\, x}{2}\right )+c_1 \,{\mathrm e}^{-\frac {\sqrt {2}\, x}{2}} \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\frac {{\mathrm e}^{2 x} \left (17 x -32\right )}{289} \]

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 88

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

\[ y(x)\to \frac {1}{289} e^{2 x} (17 x-32)+e^{-\frac {x}{\sqrt {2}}} \left (c_1 e^{\sqrt {2} x}+c_2\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+e^{-\frac {x}{\sqrt {2}}} \left (c_4 e^{\sqrt {2} x}+c_3\right ) \sin \left (\frac {x}{\sqrt {2}}\right ) \]

✓ Sympy. Time used: 0.233 (sec). Leaf size: 87

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

\[ y{\left (x \right )} = \frac {x e^{2 x}}{17} + \left (C_{1} \sin {\left (\frac {\sqrt {2} x}{2} \right )} + C_{2} \cos {\left (\frac {\sqrt {2} x}{2} \right )}\right ) e^{- \frac {\sqrt {2} x}{2}} + \left (C_{3} \sin {\left (\frac {\sqrt {2} x}{2} \right )} + C_{4} \cos {\left (\frac {\sqrt {2} x}{2} \right )}\right ) e^{\frac {\sqrt {2} x}{2}} - \frac {32 e^{2 x}}{289} \]