Mathematics MOC

Differential Equations MOC

graph LR;
  DE[differential eqn.]
  gen[general soln.]
  par[particular soln.]
  DE-->|integrate|gen-->|boundary values|par

• Classifying DEs
• Existence and uniqueness theorem for IVPs
• Boundary value problem

First order ODEs

1. Separable differential equation
2. First-order linear differential equation
3. Exact differential equation
   • Almost exact differential equations
4. Bernoulli differential equations
5. Homogenous first-order differential equation
6. System of linear ODEs

Higher order linear ODEs

A useful technique for describing properties of higher-order ODEs is to write them in terms of a Linear endomorphism

If the ODE is said to be homogenous. Given the general solution to an ODE’s related homogenous DE (called the complimentary solution), and a particular solution to the full ODE, the general solution to the full ODE is the sum of these. So any higher order ODE can be solved by Solving a homogenous ODE and Finding a particular solution.

To verify that a solution is indeed general, it is necessary to prove that solutions are linearly independent, i.e. their Wronskian determinant is zero.

Solving a homogenous ODE

1. Homogenous linear ODE with constant coëfficients
2. Cauchy-Euler differential equations
3. Converting a higher-order ODE to a system of first-order ODEs

Finding a particular solution

In practice, a variety of methods may be used to find a particular solution once has been found

• Method of undetermined coëfficients (uses an Ansatz)
• Method of annihilation
• Method of variation of parameters (similar to Reduction of order (homogenous second-order differential equation))

Link to original

Series solutions

By solving for a series solution to a DE, we can convert the differential equation into a Recurrence relation. Typically we operate on the Laurent series about a specific point, so the solution will only be valid in the neighbourhood of that point.

Given an ODE involving coëfficient functions, the radius of convergence of a series solution will be at least as large as the distance in to the nearest Singular point.

• Taylor series solutions about ordinary points

Partial differential equations

In practice, a PDE is solved by first reducing it to a system of ODEs. For this purpose there are three main techniques:

• Similarity solutions (general, in theory)
• D’Alembert solution (linear)
• Separation of variables (linear)

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