Differential Equations - created from Mind Map_1

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Classification

Ordinary (ODE) or Partial (PDE)

Order - order of the highest derivative of the dependent variable with respect to the independent variable

Linear - only if the unknown function and its derivatives appear to the power 1 Non-linear - otherwise

Homogeneous - If f(x) is a solution, so is cf(x), where c is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y Simply said, the are no constant terms in the equation They are usually recognized because the RHS is 0

Degree - the power to which one of the derivatives is raised

First Order Linear ODEs

Example: a falling object subject to linear air resistance

Solution Method

NOTE

1) Multiply through by integrating factor IF, which can be always found

2)Recall the formula for calculating the integrating factor : IF = e^

Some non-linear equations can be transformed into linear ones by change of variable

2nd and higher order - Linear ODEs with constant coefficients

Solution Method

The Wronskian

The general solution is the sum of the complementary function and the particular integral. i.e. GS = CF + PI

Complementary function

Particular Integral

General solution to the corresponding homogeneous equation

Any solution of the inhomogeneous equation

Educated Guess

The trial solutions used to find the PI are usually of the same form of the Complementary Function: most commonly constants, polynomials, sine/cosine and expontials

Series Solutions of ODEs

We can use a power series solution if the function is analytic at that point - i.e. if the function is locally given by a convergent power series

Differential Equations