85.2.9 problem 8

Internal problem ID [22436]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 1. Differential equations in general. B Exercises at page 14
Problem number : 8
Date solved : Thursday, October 02, 2025 at 08:39:24 PM
CAS classification : [[_3rd_order, _quadrature]]

\begin{align*} y^{\prime \prime \prime }&=-24 \cos \left (\frac {\pi x}{2}\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-4 \\ y \left (1\right )&=0 \\ y^{\prime }\left (1\right )&=6 \\ \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 38

ode:=diff(diff(diff(y(x),x),x),x) = -24*cos(1/2*Pi*x); 
ic:=[y(0) = -4, y(1) = 0, D(y)(1) = 6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {192 \sin \left (\frac {\pi x}{2}\right )+\left (2 x^{2}+2 x -4\right ) \pi ^{3}+192 x^{2}-384 x}{\pi ^{3}} \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 36

ode=D[y[x],{x,3}]==-24*Cos[Pi*x/2]; 
ic={y[0]==-4,y[1]==0,Derivative[1][y][1] ==6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 \left (\pi ^3 \left (x^2+x-2\right )+96 (x-2) x+96 \sin \left (\frac {\pi x}{2}\right )\right )}{\pi ^3} \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(24*cos(pi*x/2) + Derivative(y(x), (x, 3)),0) 
ics = {y(0): -4, y(1): 0, Subs(Derivative(y(x), x), x, 1): 6} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {x^{2} \left (2 \pi ^{3} + 192\right )}{\pi ^{3}} + \frac {x \left (-384 + 2 \pi ^{3}\right )}{\pi ^{3}} + \frac {192 \sin {\left (\frac {\pi x}{2} \right )}}{\pi ^{3}} - 4 \]