9.2.23 problem problem 49

Internal problem ID [957]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 5.3, Higher-Order Linear Differential Equations. Homogeneous Equations with Constant Coefficients. Page 300
Problem number : problem 49
Date solved : Tuesday, March 04, 2025 at 12:06:24 PM
CAS classification : [[_high_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=15 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 23

ode:=diff(diff(diff(diff(y(x),x),x),x),x) = diff(diff(diff(y(x),x),x),x)+diff(diff(y(x),x),x)+diff(y(x),x)+2*y(x); 
ic:=y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 15; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = {\mathrm e}^{2 x}-\frac {5 \,{\mathrm e}^{-x}}{2}-\frac {9 \sin \left (x \right )}{2}+\frac {3 \cos \left (x \right )}{2} \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 33

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

\[ y(x)\to \frac {1}{3} \left (e^x+2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )\right ) \]

✓ Sympy. Time used: 0.214 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) - Derivative(y(x), x) - Derivative(y(x), (x, 2)) - Derivative(y(x), (x, 3)) + Derivative(y(x), (x, 4)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 0, Subs(Derivative(y(x), (x, 2)), x, 0): 0, Subs(Derivative(y(x), (x, 3)), x, 0): 15} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{2 x} - \frac {9 \sin {\left (x \right )}}{2} + \frac {3 \cos {\left (x \right )}}{2} - \frac {5 e^{- x}}{2} \]