Differential Equations

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Differential Equations
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• First-order differential equations
• Concepts
• Solution of a differential equation
• Separable equations
• Homogeneous and inhomogeneous equations
• Second-order differential equations
• Concepts
• Homogeneous linear equations
• The general solution
• The particle in a 1-d box
• Special functions and the power series method
• Partial differential equations
• General solutions and an example

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Differential equations are equations that contain
derivatives

• Examples
• dx/dt kx 1st order rate process
• dx/dt k(a-x)(b-x) 2nd order rate process
• d2h/dt2 -g Body falling by gravity
• (-h2/2m)d2?/dx2 ½kx2? e?
• Wave equation for harmonic oscillator.

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General Solutions first order

• General function f(x,y,dy/dx,d2y/dx2,) 0
• By solving, or integrating, the differential
  equation our aim is to find the function(s) y(x)
  that satisfy the equation.
• dy/dx 2x? See whiteboard

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Separable equations

• Consider dy/dx f(x)/g(y)
• We can write this as
• g(y)dy f(x)dx
• This is called a separable differential equation.
• See whiteboard

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First order linear equations

• these can be written
• dy/dx p(x)y r(x)
• if r(x) 0 this is a homogeneous equation.
• if not this is an inhomogeneous equation.
• First order linear homogeneous differential
  equations are separable and are therefore easily
  soluble.
• First order linear inhomogeneous differential
  equations require just a little more work.
• See whiteboard

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Second-order differential equations. Constant
coefficients.

• d2y/dx2p(x)dy/dx q(x)y r(x)
• p(x) and q(x) are called the coefficients of the
  equation. We start with the cases where p(x) and
  r(x) are constants.

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Homogenous equations

• d2y/dx2 a.dy/dx b.y 0
• Always possible to find a solution of the form
  e?x where ? is some suitable constant.
• Example, see whiteboard.

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General solution

• See whiteboard.
• The equation y?2y 0
• Solutions y(x) c1ei?x c2e-i?x
• or
• y(x) d1 cos ?x d2 sin ?x

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Particle in a 1-d box

• Schödinger equation
• (-h2/2m)d2?(x)/dx2 V(x)?(x) E?(x)
• V(x) is zero inside box. Infinite outside box.
• So, inside the box
• (-h2/2m)d2?(x)/dx2 E?(x)
• with ?(0) ?(l)0

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Particle in a 1-d box

• (-h2/2m)d2?(x)/dx2 E?(x)
• Set ?2 (2mE)/h2
• Eqn becomes
• d2?/dx2 ?2 ? 0
• The equation y?2y 0
• Solutions y(x) c1ei?x c2e-i?x
• or
• y(x) d1 cos ?x d2 sin ?x

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Particle in a 1-d box

• d2?/dx2 ?2 ? 0
• has solution ?(x) d1 cos ?x d2 sin ?x
• Boundary conditions
• First ?(0) d1 cos 0 d2 sin 0 0
• So d1 0.
• Second ?(l) d2 sin ?l 0, which is only true
  when ?l is some multiple of ?, i.e. ?l n ? (n
  is some ve integral number, quantization!)

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Particle in a 1-d box

• So, solution
• ?(x) d2 sin (n?x/l), n 1, 2, 3, ..
• For each solution (each n) there is a given E.
• How can we get d2?

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Special functions

• Solving the equation y?2y 0, is useful in
  many problems HO, PIB, PIR
• There are other common diff. equations that pop
  up frequently enough for us to consider them
  special. e.g.
• Legendre (1-x2)y-2xyl(l1)y 0
• Hermite y-2xy2ny 0
• One way of solving these is with a power series.

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Simple illustration of the power series method

• See whiteboard.