Separation of Variables | Equations of Order One

Given the differential equation
 

$M(x, y)\,dx + N(x, y)\,dy = 0 \,\, \to \,\,$ Equation (1)

 
where $\,M\,$ and $\,N\,$ may be functions of both $\,x\,$ and $\,y\,$. If the above equation can be transformed into the form
 

$f(x)\,dx + f(y)\,dy = 0\,\, \to \,\,$ Equation (2)

 
where $\,f(x)\,$ is a function of $\,x\,$ alone and $\,f(y)\,$ is a function of $\,y\,$ alone, equation (1) is called variables separable.
 

Solution to Problem 532 | Economic Sections

Problem 532
A beam simply supported at the ends of a 25-ft span carries a uniformly distributed load of 1000 lb/ft over its entire length. Select the lightest S section that can be used if the allowable stress is 20 ksi. What is the actual maximum stress in the beam selected?

Solution to Problem 531 | Economic Sections

Problem 531
A 15-ft beam simply supported at the ends carries a concentrated load of 9000 lb at midspan. Select the lightest S section that can be employed using an allowable stress of 18 ksi. What is the actual maximum stress in the beam selected?

Solution to Problem 529 | Economic Sections

Problem 529
A 10-m beam simply supported at the ends carries a uniformly distributed load of 16 kN/m over its entire length. What is the lightest W shape beam that will not exceed a flexural stress of 120 MPa? What is the actual maximum stress in the beam selected?

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