Notes-1st order ODE pt1

# Notes-1st order ODE pt1 - What is a differential equation A...

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What is a differential equation? A differential equation is any equation containing one or more derivatives. The simplest differential equation, therefore, is just a usual integration problem y ′ = f ( t ). Comment : The solution of the above is, of course, the indefinite integral of f ( t ), y = F ( t ) + C , where F ( t ) is any antiderivative of f ( t ) and C is an arbitrary constant. Such a solution is called a general solution of the differential equation. It is really a set of infinitely many functions each differ others by one (or more) constant term and/or constant coefficients. Every differential equation, if it does have a solution, always has infinitely many functions satisfying it. All of these solutions, which differ from one another by one, or more, arbitrary constant / coefficient(s), are given by the formula of the general solution.

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Classification of Differential Equations Ordinary vs. partial differential equations An ordinary differential equation (ODE) is a differential equation with a single independent variable, so the derivative(s) it contains are all ordinary derivatives. A partial differential equation (PDE) is a differential equation with two or more independent variables, so the derivative(s) it contains are partial derivatives . Order of a differential equation The order of a differential equation is equal to the order of the highest derivative it contains. Examples : (1) y ′ + y 5 = t 2 e - t (first order ODE) (2) cos( t ) y ′ - sin( t ) y = 3 t cos( t ) (first order ODE) (3) y ″ - 3 y ′ + 2 y = e 2 t cos(5 t ) (second order ODE) (4) y (4) + ( y ) 30 = 0 (fourth order ODE) (5) u xx = 4 u tt + u t (second order PDE) (6) y (3) - ( y y ′) + 2 y = 4 e 7 t (third order ODE)
: An n -th order ordinary differential equation is called linear if it can be written in the form: y ( n ) = a n ±1 ( t ) y ( n ±1) + a n ±2 ( t ) y ( n ±2) + … + a 1 ( t ) y ′ + a 0 ( t ) y + g ( t ) . Where the functions a ’s and g are any functions of the independent variable, t in this instance. Note that the independent variable could appear in any shape or form in the equation, but the dependent variable, y , and its derivatives can only appear alone, in the first power, not in a denominator or inside another (transcendental) function. In other words, the right-hand side of the equation above must be a linear function of the dependent variable y and its derivatives. Otherwise, the equation said to be nonlinear . In the examples above, (2) and (3) are linear equations, while (1), (4) and (6) are nonlinear. (5) is a linear partial differential equation, as each of the partial derivatives appears alone in the first power. The next example looks similar to (3), but it is a (second order) nonlinear equation, instead. Why? (7)

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Notes-1st order ODE pt1 - What is a differential equation A...

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