Transcript Chapter 8: Ordinary Differential Equations I. General

Ch. 8- Ordinary Differential Equations > General
Chapter 8: Ordinary Differential Equations
I. General
•
A linear ODE is of the
form:3
2
ao y  a1 dy
 a2 ddxy2  a3 ddxy3 
dx
•
b
An nth order ODE has a solution containing n arbitrary constants
ex: d  cos
dy
(1st order)
 y  sin  C1
ex:
d 2y
d
 cos 
(2nd order)
 y   cos   C1  C2 (2 constants)
Ch. 8- Ordinary Differential Equations > General
• Three really common ODE’s:
1)
  y  y  Ae x (whether λ is positive or negative)
dy
dx
ex: y   -3y  y  Ae 3 x
2)
d 2y
dx
2
  2 y  y  Ae   x  Be  x
check this: y    Ae   x  Be  x
y    2 Ae   x   2Be  x   2 y
3)
d 2y
dx
2
  2 y  y  A cos  x  B sin  x
check this: y    A sin  x  B cos  x
y    2 A cos  x   2B sin  x   2 y
Ch. 8- Ordinary Differential Equations > General
• How do we solve for the constants?
→ In general, any constant works.
→ But many problems have additional constants (boundary conditions) and in
this case, the particular solution involves specific values of the constants
that satisfy the boundary condition.
ex:
for t<0, the switch is open and the capacitor is uncharged.
at t=0, shut switch
Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations
II. Separable Ordinary Differential Equations
A separable ODE is one in which you can separate all y-terms on the left hand side of
the equation and all the x-terms on the right hand side of the equation.
ex: xy’=y
We can solve separable ordinary differential equations by separating the variables
and then just integrating both sides
Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations
ex: xy’=y subject to the boundary condition y=3 when x=2
Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations
ex: Rate at which bacteria grow in culture is proportional to the present.
Say there are no bacteria at t=0.
dN
 aN
dt
subject to boundary condition N(t=0)=No
Ch. 8- Ordinary Differential Equations > Separable Ordinary Differential Equations
 2 ( x, t )



V
(
x
,
t
)

(
x
,
t
)

i
ex: Schroedinger’s Equation:
2m x 2
t
2
solve for the wave function ( x, t )
if V(x,t) is only a function of x, e.g. V(x), then schroedinger’s equation is separable.
 2 ( x, t )



V
(
x
)

(
x
,
t
)

i
2m x 2
t
2
Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs
III. Linear First-Order Ordinary Differential Equations
Definition: a linear first-order ordinary differential equation can be written in the form:
y’+Py=Q where P and Q are functions of x
the solution to this is:
y  e  I  Qe I dx  Ce  I
Check:
Ch. 8- Ordinary Differential Equations > Linear First-Order ODEs
ex: 2 xy   y  2 x
5
2
Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation
IV. Second Order Linear Homogeneous Equation
A second order linear homogeneous equation has the form:
d 2y
dy
a2 2  a1
 a0 y  0
dx
dx
where a , a , a are constants
To solve such an equation:
let D 
d
dx
d 2y
dy

c
 c2 y  0
1
dx 2
dx
 (D 2  K1D  K 2 )y  0
 (D  a )(D  b )y  0
C1eax  C2e bx if a  b
y 
ax
if a  b
 ( Ax  B )e
2
1
0
Ch. 8- Ordinary Differential Equations > Second-Order Linear Homogeneous Equation
ex: y’’+y’-2y=0
ex: Harmonic Oscillator