Solutions of Differential Equations: A Comprehensive Guide for Engineering Students, Cheat Sheet of Design

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LESSON 2: SOLUTIONS

OF DIFFERENTIAL

EQUATIONS

ENGR. PRETTY JOY V. CENTENO

INSTRUCTOR

• It is an equation that is defined as a function or a set of

functions free of derivatives or differentials expressing

the functional relationship between the dependent and

independent variables, it may be general or particular.

The integration will be the major step in solving the

solutions of a differential equation.

• A solution to a DE is a function (let’s say 𝑦 = 𝑓 𝑥( ))

that satisfies the DE when f and its derivatives are

substituted into the equation.

SOLUTION TO A DE

2 TYPES:

1. GENERAL SOLUTION

2. PARTICULAR

SOLUTION

• A relation between the variables that involves n essential arbitrary constants is called a general solution or primitive. The n constants are called essentials if they cannot be replaced by a smaller number of constants. This is given by the equation, � �( 𝑥 𝑦 𝐶, , 1, 𝐶2, ⋯ 𝐶𝑛, −1, 𝐶𝑛) = 0 where: 𝐶 1, 𝐶2, ⋯ 𝐶𝑛, −1, 𝐶𝑛 are the arbitrary constants. GENERAL SOLUTION

IN SHORT:

- Sa General Solution kukunin po lang yung integration, pero sa Particular Solution kukunin mo ang value ng C and substitute sa General Solution

A primitive involving n essential arbitrary constants will give rise to a DE, of order n, free of arbitrary constants. This equation is obtained by eliminating the n constants between the (n+1) equations consisting of the primitive and the n equations obtained by differentiating the primitive n times with respect to the independent variable. FINDING THE DE BY ELIMINATION OF ARBITRARY CONSTANTS

Properties:

o The order of the differential equation is equal to the

number of arbitrary constants in the given relation.

o The differential equation is consistent with the

relation.

o The differential equation is free from arbitrary

constants.