,
Integration: Differential Equations
 

[ introduction ][ variables separable ][ linear ]

 

 

 

 

Introduction

All equations with derivatives of a variable w.r.t. another are called 'differential equations'. A first order differential equation contains a first derivative eg dy/dx.

It might not be appreciated, but ALL integrals are derived from original 'first-order' differential equations.

differential equations theory#1

Example:

integration of differential eqs. theory #2

 

back to top

 

First Order with 'variables separable'

Solution is by collecting all the 'y' terms on one side, all the 'x' terms on the other and integrating each expression independently.

separable variables #1

 

Example #1

separable differential equations problem#1

Note how the constant of integration C changes its value.

 

Example #2

separated variable problem#2

 

back to top

 

First Order 'linear' differential equations

By definition 'linear' differential equation have the form:

linear differential equation#1

Dividing by f(x) to make the coefficient of dy/dx equal to '1', the equation becomes:

linear differential equation#2

(where P and Q are functions of x, and only x)

The key to solving these types of problem is to choose a multiplying factor(sometimes called an 'integrating factor') to make the LHS of the equation appear like a result from the Product Rule.

product rule

Example

differential equation problem#1

 

 

 

back to top

 

your stop for the best in math, science & programming tutorials on the Net revision help to get a better result incremental success advanced physics for secondary/high school, including much in-depth content common to first year university courses your one stop for the best in math, science and programming tuition revision help for a better result incremental success advanced physics for high school/secondary and 1st year university fast-track learning for everyone

[ PURE MATHS ][ MECHANICS ][ STATISTICS ]

VIDEO

parametric differen.
equation of a tangent
equation of a normal
rate of change prob.1
rate of change prob.2
the Chain Rule
Chain Rule probs. #1
Chain Rule probs. #2
Chain Rule probs. #3
the Product Rule
parametric eqs.prob#1
parametric eqs.prob#2
intro. to integration
integration by parts 1
integration by parts 2
area under a curve
volumes of revolution
area between curves
Binomial Theorem
Bin. Theorem problems
Trig. Identities
Half Angle Formula
Double Angle Formula
Vectors,mod,resultant
Vector problems
Vector problems in 3D
MORE . . .
 

INTERACTIVE

 
derivative formula
tangents & normals
inverse functions
MORE . . .
 

EXAM PAPERS(.pdf)

 
Edxl C1 Pure specimen
Edxl C1 Pure answers
Edxl C2 Pure specimen
Edxl C2 Pure answers
Edxl C3 Pure specimen
Edxl C3 Pure answers
Edxl C4 Pure specimen
Edxl C4 Pure answers
MORE . . .

TOPIC NOTES(.pdf)

the derivative formula
tangents & normals
maxima & minima
chain rule
diffn.exponentials,logs
diffn.trigonometric fns.
product rule
quotient rule
parametric equations
implicit equations
differential equations
integration formula
int.'by substitution'
int.'by parts'
algebraic fractions
definite integrals
areas under curves
volumes of revolution
Trapezium Rule
integ. diff. equations
radians
sine, cosine, tangent
sec, cosec, cotan
Sine & Cosine Rules
Pythagorean ID's
compound angles
Sigma Notation
arithmetic progression
geometic progression
line between points
more straight lines
parametric equations
circles & ellipses
MORE . . .
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Google