7.10.38 problem 38

Internal problem ID [308]
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 2.3 (Homogeneous equations with constant coefficients). Problems at page 134
Problem number : 38
Date solved : Tuesday, March 04, 2025 at 11:07:36 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-5 y^{\prime \prime }+100 y^{\prime }-500 y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{5 x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=10\\ y^{\prime \prime }\left (0\right )&=250 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 17

ode:=diff(diff(diff(y(x),x),x),x)-5*diff(diff(y(x),x),x)+100*diff(y(x),x)-500*y(x) = 0; 
ic:=y(0) = 0, D(y)(0) = 10, (D@@2)(y)(0) = 250; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = 2 \,{\mathrm e}^{5 x}-2 \cos \left (10 x \right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 19

ode=D[y[x],{x,3}]-5*D[y[x],{x,2}]+100*D[y[x],x]-500*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==10,Derivative[2][y][0] ==250}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to 2 \left (e^{5 x}-\cos (10 x)\right ) \]

✓ Sympy. Time used: 0.193 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-500*y(x) + 100*Derivative(y(x), x) - 5*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 10, Subs(Derivative(y(x), (x, 2)), x, 0): 250} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = 2 e^{5 x} - 2 \cos {\left (10 x \right )} \]