Engng Math 214 Theory of LDE

LDE of n-th order

a_n(x)yⁿ + a_n-1(x)y^n-1 + … + a₀(x)y = g(x)

2 problem types

• IV: values given at single point (x₀)

• BV: values given at two points (x₀, x)

Superposition principle

If y = f₁, y = f₂, y = f_k are solutions of homogeneous LDE, then y = c₁f₁ + c₂f₂ + … + c_kf_k is also a solution.

Linearly dependant/independant solutions

• Linearly dependant if c₁, c₂, c_n are all ≠ 0 such that c₁f₁ + c₂f₂ + … + c_kf_k = 0 on interval I.

  • Also 2 solutions LD if dividing them by each other gives a constant.

• Linearly independant if all c₁, c₂, c_n = 0 on interval I.

The Wronskian

For f₁, f₂, …, f_k on interval I, each f_k is differential at least k-1 times:

• Linearly independant if W not equal to 0

• Linearly dependant if W ≡ 0

Fundamental set

Fundamental set if:

• number of solutions = n

• set is linearly independant

Can be written in form {y₁, y₂, …, y_n}

General solution

If you have fundamental set of solutions of LDE, then general solution is given by:

• y = y_p + c₁y₁ + c₁y₂ + … + c_ny_n

where y_p is a solution of LDE

Verify general/particular solution question

y is given, so find y’ & y’’ then sub into DE & make LHS = RHS to prove that y is general/particular solution.