MATH2310 – DE Lecture 3
Linear ODE with constant coefficients:
These have the general form:
dn y d n−1 y
a n n +a n−1 n−1 +…+a 0 y =f ( x)
dx dx
Where c n , c n−1 , … , c 0 are real or complex numbers.
Linear independence:
A collection of functions is said to be linearly independent if they cannot be
expressed as a combination of each other.
A set of functions is linearly independent if their Wronskian is not zero.
Wronskians:
y1 y2 ⋯ yn
(n−1) (n−1)
W ( y1 , … , yn)= y1 '
y2
'
… y n y2
'
¿ … ¿ yn ¿
⋮ ⋮ ¿ ¿
If the Wronskian is non-zero on an interval I , then the functions y 1 , … , y n are
independent on I .
Sarrus rule:
Fundamental set of solutions:
Linear ODE with constant coefficients:
These have the general form:
dn y d n−1 y
a n n +a n−1 n−1 +…+a 0 y =f ( x)
dx dx
Where c n , c n−1 , … , c 0 are real or complex numbers.
Linear independence:
A collection of functions is said to be linearly independent if they cannot be
expressed as a combination of each other.
A set of functions is linearly independent if their Wronskian is not zero.
Wronskians:
y1 y2 ⋯ yn
(n−1) (n−1)
W ( y1 , … , yn)= y1 '
y2
'
… y n y2
'
¿ … ¿ yn ¿
⋮ ⋮ ¿ ¿
If the Wronskian is non-zero on an interval I , then the functions y 1 , … , y n are
independent on I .
Sarrus rule:
Fundamental set of solutions:
