85.52.2 problem 2

Internal problem ID [22813]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. C Exercises at page 195
Problem number : 2
Date solved : Thursday, October 02, 2025 at 09:14:52 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Maple. Time used: 0.087 (sec). Leaf size: 31

ode:=diff(diff(y(x),x),x)+y(x) = piecewise(0 <= x and x <= Pi,x,Pi < x,0); 
ic:=[y(0) = 0, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left \{\begin {array}{cc} 0 & x <0 \\ x -\sin \left (x \right ) & x <\pi \\ -\cos \left (x \right ) \pi -2 \sin \left (x \right ) & \pi \le x \end {array}\right . \]

✓ Mathematica. Time used: 0.04 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to \theta (\pi -t) \theta (t) (x-\sin (x)) \end{align*}

✓ Sympy. Time used: 0.191 (sec). Leaf size: 10

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

\[ y{\left (x \right )} = \begin {cases} 1 & \text {for}\: x \geq 0 \wedge x \leq \pi \\0 & \text {for}\: x > \pi \\\text {NaN} & \text {otherwise} \end {cases} - \cos {\left (x \right )} \]