Differential Equations

Differential Equation

It is one in which involves one or more independent variables, a dependent variable and one or more of their differential coefficients.

ORDINARY DE

the unknown function depends on only one independent variable

PARTIAL DE

the unknown function depends on two or more independent variables

dependent variable

dy in dy/dx (numerator)

Independent Variable

dx in dy/dx (denominator)

Differential Coefficients

dy/dx
- another name for the derivative of a function

- It measures how fast a function is changing with respect to its variable at a particular point.

ORDER

The highest ordered derivative appearing in a differential equation.
*more important than degree

DEGREE

The exponent of the highest ordered derivative.

*must be a whole number so fix the exponent first if it is in fraction form

DE OF FAMILY OF CURVES

A family of curves on a plane is usually defined by an equation containing one or more parameters together with the coordinates of a point on the plane.

Number of parameters (arbitrary constant)

The one that distinguishes one form from another in a family of curves

Arbitrary constant

• No. of times to derive y

• *DE must have no parameters

If “Find the differential equation”

Derive y

If “Find the solution to the differential equation”

Integrate the given DE

GENERAL SOLUTION

set of all possible solutions, which includes the particular and singular solutions.

PARTICULAR SOLUTION

solution obtained from the general solution by assigning particular values to the arbitrary constants.

Order of DE

Number of times to integrate

VARIABLE SEPARABLE

Possible to re-arrange the terms of the differential equation in two groups, each containing only one variable, the variables are said to be separable.

HOMOGENEOUS

• A differential equation where, x and y are replaced by λx and λy, it will be written in the form:

• To solve, put y = ux then dy = udx + xdu and the differential equation reduces to variable separable.

• Short cut in checking: same degree both sides

EXACT DE

A differential equation which is the result of a simple differentiation.

Test for exactness

*MAY NAX

INTEGRATING FACTOR for EXACT DE

A given D.E. may not be integrable. But it may become integrable when it is multiplied by a function. Such function is called

Advantage of the solution of Exact DE

Even if the equation is not exact, there is a way for it to be exact by incorporation of an Integrating Factor (compared to homogenous that doesn’t have another way)

To solve for P in the Integrating Factor for Exact DE

LINEAR DE

To be linear, the dependent variable should be linear.
There can be any sort of complicated function of x in the equation. Thus a linear equation can be written as

Conditions for LINEAR DE

Linear in Y

yes Pdx, QeSPdx, dxC

Linear in X

xes Pdy, QeSPdy, dyC

General equation for the solution of Linear DE (at x)

BERNOULLI'S DE

Some equations not in the linear form may be reduced to the linear form by a suitable substitution.

Solution to Bernoulli’s DE

n = 0, linear
n = 1, homogenous
n ≠ 1 or 0, Bernoulli

conditions on how to choose appropriate solution to DE

HIGHER ORDER LINEAR DIFFERENTIAL EQUATION

An ordinary differential equation of order n is called linear if it may be written in the form:

DIFFERENTIAL OPERATOR

When placed before a function of x, it means that the function is to be differentiated.

Case 1: Overdamped

Roots: Real & Distinct

Case 2: Critically Damped

Roots: Real & Equal

Case 3: Underdamped

Roots: Complex and Conjugate

Case 1: Overdamped solution

Case 2: Critically Damped solution

Case 3: Underdamped

HOMOGENEOUS LINEAR DE solution

NON-HOMOGENEOUS LINEAR DE solution

• Algebraic

• Exponential

• Trigonometric

To what equations can we apply non-homogenous linear DE

ORTHOGONAL TRAJECTORIES

• Family of curves G(x,y,x) intersecting another family of curves F(x,y,c) @ right angles (90°)

• Application of lower math (2 lines only) to higher math (finding 90° @ all points in the curve)

Solution to Orthogonal Trajectories

1. Differentiate to get slope

2. Negate and reciprocate the slope

3. Integrate

NATURAL GROWTH

"The rate of change of y is proportional to the present value of y."

Population Growth

Unrestricted population growth

Radioactive Decay

Radioactive Decay

Bank Accounts

Continuous Compounding

SHIFTED NATURAL GROWTH

"The rate of change of y is proportional to the value of y along with some other constant growth/decay rate."

NEWTON'S LAW OF COOLING

"The rate of change of temperature of an object is proportional to the difference between the object and surrounding temperature."

MIXING PROBLEMS

• rate in L/min

• concentration density in kg/L, g/L
  

Solving for Co (output concentration) in Mixing Problems

Concentration or Density

Amount of substance / volume; mass / volume

Case 1 in Mixing Problems

Ri = Ro
Solved using either Exact or Linear

Case 2 in Mixing Problems

Ri ≠ Ro

Case 3 in Mixing Problems

Ro = 0

LIMITED EXPONENTIAL GROWTH

“The rate of change of P is proportional to the difference of the maximum and present value of P.”

Shortcut in solving for Shifted Exponential Growth

Logistic Growth

“The rate of change of population is proportional to the present value of P and to the allowable additional value of P.”