Engineering Mat...

The order and degree of the differential equation \({\left( {\frac{{{d^4}y}}{{d{x^4}}}} \right)^{\frac{1}{2}}} = {\left[ {1 + {{\left( {\frac{{{d^2}y}}{{d{x^2}}}} \right)}^2}} \right]^{\frac{1}{3}}}\) respectively are

Consider the following four differential equations:

EQ 1:  \(\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - {\rm{y}} = {\rm{\pi }}\)

EQ 2: \({\rm{\;}}{\left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right)^2} + 3{\rm{y}} = 36\)

EQ 3:  \(\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}} + 5\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + 6{\rm{y}} = {\rm{\;\;}}0\)

EQ 4: \({\rm{\;}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + {\rm{y}} = {\rm{tany}}\)

What is the integrating factor of the following differential equation if any?

\(\cos {\rm{x}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + {\rm{y}}\tan {\rm{x}} = {\cos ^3}{\rm{x}}\)

If roots of the auxiliary equation of \(\frac{{{d^2}y}}{{d{x^2}}} + a\frac{{dy}}{{dx}} + by = 0\) are real and equal, the general solution of the differential equation is

The solutions of Differential equation \(\left( {x{y^2}-{{\rm{e}}^{\frac{1}{{{{\rm{x}}^3}}}}}} \right)dx - {x^2}y\;dy = 0\) is

What is the particular integral of the differential equation: \(\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}} - \frac{{{\rm{dy}}}}{{{\rm{dx}}}} - 20{\rm{y}} = \sinh 5{\rm{x}}\)?

The solution of differential equation

Solution of\(\;\frac{{{d^2}y}}{{d{x^2}}} + \frac{{2dy}}{{dx}} + 17y = 0\); y(o) = 2, y’ \(\left( {\frac{\pi }{2}} \right) = 0\)

Structural analysis of a theoretical beam which is subjected to an irregular loading system shows that the deflection of the beam can be best represented by the differential equation \({\rm{\;}}\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}} + \frac{2}{{\rm{x}}}\frac{{{\rm{dy}}}}{{{\rm{dx}}}} - 6\frac{{\rm{y}}}{{{{\rm{x}}^2}}}{\rm{\;}} = 7{{\rm{x}}^2}\) where x is the distance from a certain origin point and y is deflection. Two constraint conditions enable deflection to be zero at x = 1 and to be 1 unit at x = -1. What will be the deflection (in unit corrected up to three decimal points) at x =2?