71.12.2 problem 2

Internal problem ID [14439]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 4. N-th Order Linear Differential Equations. Exercises 4.5, page 221
Problem number : 2
Date solved : Saturday, February 22, 2025 at 03:48:33 PM
CAS classification : [[_high_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime \prime }-2 y^{\prime \prime \prime }+2 y^{\prime }-y&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1\\ y^{\prime \prime }\left (0\right )&=-3\\ y^{\prime \prime \prime }\left (0\right )&=3 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 24

ode:=diff(diff(diff(diff(y(x),x),x),x),x)-2*diff(diff(diff(y(x),x),x),x)+2*diff(y(x),x)-y(x) = 0; 
ic:=y(0) = 1, D(y)(0) = -1, (D@@2)(y)(0) = -3, (D@@3)(y)(0) = 3; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = -{\mathrm e}^{-x}+\left (2 x^{2}-4 x +2\right ) {\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 25

ode=D[y[x],{x,4}]-2*D[y[x],{x,3}]+2*D[y[x],x]-y[x]==0; 
ic={y[0]==1,Derivative[1][y][0] ==-1,Derivative[2][y][0] ==-3,Derivative[3][y][0]==3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-x} \left (2 e^{2 x} (x-1)^2-1\right ) \]

✓ Sympy. Time used: 0.263 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-y(x) + 2*Derivative(y(x), x) - 2*Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -1, Subs(Derivative(y(x), (x, 2)), x, 0): -3, Subs(Derivative(y(x), (x, 3)), x, 0): 3} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (x \left (2 x - 4\right ) + 2\right ) e^{x} - e^{- x} \]