Question

​​​​​​Question 1. Substitution of given form of solution and hyperbolic functions.

Question 1. Substitution of given form of solution and hyperbolic functions. The non-linear ordinary differential equation describing the smooth shape of a structural arch of constant thickness in mechanical equilibrium under its own weight per unit length w, and a horizontal compressive force T, is Here k = is a known constant and y(x) is the vertical height of the arch at position x, the horizontal distance from a given reference point. (a) Using hyperbolic function properties (see Appendix A.2 of the printed notes), show that there is a solution satisfying ()sinh(kr C) Here C is a yet-to-be-determined constant. (b) Hence find a solution y(x). (As well as C, it will contain another undetermined constant.)

Answer 1

Solution:

we have the equation,

dy dr

we assume y 2

By assuming y-z we can represent the above as a first order equation:

$z'=k\sqrt {1 + {z^2}}$

$\mathrm{d}z=k\sqrt {1 + {z^2}}\mathrm{d}x$

dz

$\ln \left( {z + \sqrt {1 + {z^2}} } \right)=kx+C$

$z + \sqrt {1 + {z^2}} =e^{kx+C}$

$\text{Multiplying both sides of the equation by the conjugent expression } z - \sqrt {1 + {z^2}}$

$\left( z + \sqrt {1 + {z^2}} \right)\cdot \left( z - \sqrt {1 + {z^2}} \right) = \left( z - \sqrt {1 + {z^2}} \right)e^{kx+C}$

$z - \sqrt {1 + {z^2}} =-e^{-kx-C}$

$2z=e^{kx+C}-e^{-kx-C}$

$z=\frac{e^{kx+C}-e^{-kx-C}}{2}$

$z=\sinh \left (kx+C \right )$

$\frac{\mathrm{d} y}{\mathrm{d} x}=\sinh \left ( kx+C \right )$

$\text{b)}$

$\int \mathrm{d} y=\int \sinh \left ( kx+C \right )\mathrm{d} x$

cosh (kx C), + C1 y(x)-