Integration: Differential Equations

[ introduction ][ variables separable ][ linear ]






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


integration of differential eqs. theory #2


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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


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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


differential equation problem#1




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parametric differen.
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equation of a normal
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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
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Half Angle Formula
Double Angle Formula
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the derivative formula
tangents & normals
maxima & minima
chain rule
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
sine, cosine, tangent
sec, cosec, cotan
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circles & ellipses
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