71.15.3 problem 4 (c)

Internal problem ID [14474]
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 (c)
Date solved : Monday, March 31, 2025 at 12:28:01 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }&=\left \{\begin {array}{cc} 0 & 0\le x <1 \\ \left (x -1\right )^{2} & 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: 0.199 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x) = piecewise(0 <= x and x < 1,0,1 <= x,(x-1)^2); 
ic:=y(0) = 1, D(y)(0) = 0; 
dsolve([ode,ic],y(x),method='laplace');
 

\[ y = \left \{\begin {array}{cc} 1 & x <1 \\ \frac {7}{8} & x =1 \\ \frac {25}{24}+\frac {{\mathrm e}^{2 x -2}}{8}-\frac {x^{3}}{6}+\frac {x^{2}}{4}-\frac {x}{4} & 1<x \end {array}\right . \]

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 40

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

\[ y(x)\to \begin {array}{cc} \{ & \begin {array}{cc} 1 & x\leq 1 \\ \frac {1}{24} \left (-4 x^3+6 x^2-6 x+3 e^{2 x-2}+25\right ) & \text {True} \\ \end {array} \\ \end {array} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-Piecewise((0, (x >= 0) & (x < 1)), ((x - 1)**2, x >= 1)) - 2*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)