71.15.7 problem 4 (h)

Internal problem ID [14478]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 5. The Laplace Transform Method. Exercises 5.4, page 265
Problem number : 4 (h)
Date solved : Monday, March 31, 2025 at 12:28:10 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-4 y^{\prime }+5 y&=\left \{\begin {array}{cc} x & 0\le x <1 \\ 1 & 1\le x \end {array}\right . \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 0.360 (sec). Leaf size: 85

ode:=diff(diff(y(x),x),x)-4*diff(y(x),x)+5*y(x) = piecewise(0 <= x and x < 1,x,1 <= x,1); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = \frac {\left (\left \{\begin {array}{cc} 4+{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right )+5 x & x <1 \\ 10+{\mathrm e}^{2} \left (21 \cos \left (1\right )-47 \sin \left (1\right )\right ) & x =1 \\ 5+{\mathrm e}^{2 x} \left (21 \cos \left (x \right )-47 \sin \left (x \right )\right )+\left (-3 \sin \left (x -1\right )+4 \cos \left (x -1\right )\right ) {\mathrm e}^{-2+2 x} & 1<x \end {array}\right .\right )}{25} \]

✓ Mathematica. Time used: 0.049 (sec). Leaf size: 119

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

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} e^{2 x} (\cos (x)-2 \sin (x)) & x\leq 0 \\ \frac {1}{25} \left (5 x+21 e^{2 x} \cos (x)-47 e^{2 x} \sin (x)+4\right ) & 0<x\leq 1 \\ \frac {4 e^{2 x} \cos (1-x)+21 e^{2 x+2} \cos (x)+3 e^{2 x} \sin (1-x)-47 e^{2 x+2} \sin (x)+5 e^2}{25 e^2} & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Piecewise((x, (x >= 0) & (x < 1)), (1, x >= 1)) + 5*y(x) - 4*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)