65.14.8 problem 14

Internal problem ID [15829]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.3, page 255
Problem number : 14
Date solved : Thursday, October 02, 2025 at 10:28:23 AM
CAS classification : [[_3rd_order, _missing_x]]

\begin{align*} y^{\prime \prime \prime }-y^{\prime \prime }+4 y^{\prime }-4 y&=0 \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=5 \\ y^{\prime \prime }\left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 19

ode:=diff(diff(diff(y(x),x),x),x)-diff(diff(y(x),x),x)+4*diff(y(x),x)-4*y(x) = 0; 
ic:=[y(0) = 0, D(y)(0) = 5, (D@@2)(y)(0) = 5]; 
dsolve([ode,op(ic)],y(x),method='laplace');
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 21

ode=D[y[x],{x,3}]-D[y[x],{x,2}]+4*D[y[x],x]-4*y[x]==0; 
ic={y[0]==0,Derivative[1][y][0] ==5,Derivative[2][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x+2 \sin (2 x)-\cos (2 x) \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*y(x) + 4*Derivative(y(x), x) - Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 5, Subs(Derivative(y(x), (x, 2)), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

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