62.27.7 problem Ex 7

Internal problem ID [12866]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 50. Method of undetermined coefficients. Page 107
Problem number : Ex 7
Date solved : Wednesday, March 05, 2025 at 08:48:54 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 38

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

\[ y = 4+\left (c_4 x +c_{2} \right ) {\mathrm e}^{-x}+\frac {\left (3+2 x^{2}+4 \left (-1+4 c_3 \right ) x +16 c_{1} \right ) {\mathrm e}^{x}}{16} \]

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 124

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

\[ y(x)\to \frac {1}{8} e^{-x} \left (8 \int _1^x-\frac {1}{4} e^{K[1]} \left (4+e^{K[1]}\right ) (K[1]-1)dK[1]+8 e^{2 x} \int _1^x-\frac {1}{4} e^{-K[2]} \left (4+e^{K[2]}\right ) (K[2]+1)dK[2]+2 e^{2 x} x^2+e^{2 x} x+16 x+8 c_2 x+8 c_4 e^{2 x} x+8 c_3 e^{2 x}+8 c_1\right ) \]

✓ Sympy. Time used: 0.136 (sec). Leaf size: 24

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

\[ y{\left (x \right )} = \left (C_{1} + C_{2} x\right ) e^{- x} + \left (C_{3} + x \left (C_{4} + \frac {x}{8}\right )\right ) e^{x} + 4 \]