75.5.16 problem 16

Internal problem ID [19964]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter IV. Ordinary linear differential equations with constant coefficients. Exercises at page 58
Problem number : 16
Date solved : Thursday, October 02, 2025 at 05:04:31 PM
CAS classification : [[_2nd_order, _quadrature]]

\begin{align*} e y^{\prime \prime }&=\frac {w \left (\frac {L^{2}}{4}-x^{2}\right )}{2} \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 28

ode:=e*diff(diff(y(x),x),x) = 1/2*w*(1/4*L^2-x^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {w \left (3 L^{2}-4 x^{2}\right )^{2}}{384 e}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 34

ode=e*D[y[x],{x,2}]==w/2*(L^2/4-x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {w x^2 \left (3 L^2-2 x^2\right )}{48 e}+c_2 x+c_1 \end{align*}

✓ Sympy. Time used: 0.055 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
L = symbols("L") 
e = symbols("e") 
w = symbols("w") 
y = Function("y") 
ode = Eq(e*Derivative(y(x), (x, 2)) - w*(L**2/4 - x**2)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} + C_{2} x + \frac {L^{2} w x^{2}}{16 e} - \frac {w x^{4}}{24 e} \]