Notes-2nd order ODE pt1

Notes-2nd order ODE pt1 - Second Order Linear Differential...

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Second Order Linear Differential Equations Second order linear equations with constant coefficients; Fundamental solutions; Wronskian; Existence and Uniqueness of solutions; the characteristic equation; solutions of homogeneous linear equations; reduction of order In this chapter we will study ordinary differential equations of the standard form below, known as the second order linear equations : y ″ + p ( t ) y ′ + q ( t ) y = g ( t ). Homogeneous Equations : If g ( t ) = 0, then the equation above becomes y ″ + p ( t ) y ′ + q ( t ) y = 0. It is called a homogeneous equation. Otherwise, the equation is nonhomogeneous (or inhomogeneous ).
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Second Order Linear Homogeneous Differential Equations with Constant Coefficients For the most part, we will only learn how to solve second order linear equation with constant coefficients (that is, when p ( t ) and q ( t ) are constants). Since a homogeneous equation is easier to solve compares to its nonhomogeneous counterpart, we start with second order linear homogeneous equations that contain constant coefficients only: a y ″ + b y ′ + c y = 0. Where a , b , and c are constants, a ≠ 0. A very simple instance of such type of equations is y ″ - y = 0 . The equation’s solution is any function satisfying the equality y ″ = y . Obviously y 1 = e t is a solution, and so is any constant multiple of it, C 1 e t . Not as obvious, but still easy to see, is that y 2 = e -t is another solution (and so is any function of the form C 2 e -t ). It can be easily verified that any function of the form y = C 1 e t + C 2 e -t will satisfy the equation. In fact, this is the general solution of the above differential equation. Comment : Unlike first order equations we have seen previously, the general solution of a second order equation has two arbitrary coefficients.
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Principle of Superposition : If y 1 and y 2 are any two solutions of the homogeneous equation y ″ + p ( t ) y ′ + q ( t ) y = 0. Then any function of the form y = C 1 y 1 + C 2 y 2 is also a solution of the equation, for any pair of constants C 1 and C 2 . That is, for a homogeneous linear equation, any multiple of a solution is again a solution; any sum/difference of 2 solutions is again a solution; and the sum / difference of the multiples of any 2 solutions is again a solution. (This principle holds true for a homogeneous linear equation of any order; it is not a property limited only to a second order equation. It, however, never holds for solutions of a nonhomogeneous linear equation.) Note : However, while the general solution of y ″ + p ( t ) y ′ + q ( t ) y = 0 will always be in the form of C 1 y 1 + C 2 y 2 , where y 1 and y 2 are some solutions of the equation, the converse is not always true. Not every pair of solutions y 1 and y 2 could be used to give a general solution in the form y = C 1 y 1 + C 2 y 2 . We shall see shortly the exact condition that
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Notes-2nd order ODE pt1 - Second Order Linear Differential...

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