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Differential Equations

- 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

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.

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

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

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

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.

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.

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

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

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

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!)

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?

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.

Simple illustration of the power series method

- See whiteboard.

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