# Solution to Problem 536 | Economic Sections

**Problem 536**

A simply supported beam 10 m long carries a uniformly distributed load of 20 kN/m over its entire length and a concentrated load of 40 kN at midspan. If the allowable stress is 120 MPa, determine the lightest W shape beam that can be used.

# Solution to Problem 535 | Economic Sections

**Problem 535**

A simply supported beam 24 ft long carries a uniformly distributed load of 2000 lb/ft over its entire length and a concentrated load of 12 kips at 8 ft from left end. If the allowable stress is 18 ksi, select the lightest suitable W shape. What is the actual maximum stress in the selected beam?

# Solution to Problem 534 | Economic Sections

**Problem 534**

Repeat Prob. 533 if the uniformly distributed load is changed to 5000 lb/ft.

# Problem 01 | Separation of Variables

**Problem 01**

$\dfrac{dr}{dt} = -4rt$, when $t = 0$, $r = r_o$

**Solution 01**

$\dfrac{dr}{dt} = -4rt$

$\dfrac{dr}{r} = -4t\,dt$

# Linear Equations | Equations of Order One

**Linear Equations of Order One**

Linear equation of order one is in the form

The general solution of equation in this form is

# Exact Equations | Equations of Order One

The differential equation

is an exact equation if

# Homogeneous Functions | Equations of Order One

If the function f(x, y) remains unchanged after replacing x by kx and y by ky, where k is a constant term, then f(x, y) is called a **homogeneous function**. A differential equation

is homogeneous in x and y if M and N are homogeneous functions of the same degree in x and y.

# Derivation of Formula for Sum of Years Digit Method (SYD)

The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:

**Depreciation Charge:**

**Total depreciation at any time m**

Where:

FC = first cost

SV = salvage value

n = economic life (in years)

m = any time before n (in years)

SYD = sum of years digit = 1 + 2 + ... + n = n(1 + n)/2

# Separation of Variables | Equations of Order One

Given the differential equation

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

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