Linear differential equation d2y dx2 + a dy dx + by = f(x), Slides of Differential Equations

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Linear differential equation

d^2 y dx^2

• a

dy dx

• by = f (x)

Linear differential equation

d^2 y dx^2

• a

dy dx

• by = f (x)

Examples: d^2 y dx^2 + 2^

dy dx +^ y^ =^ e

x

d^2 y dx^2

• x^2

dy dx

• y = 0

The equation of pendulum motion is nonlinear

d^2 θ dt^2

g ℓ

sin θ = 0

First order linear differential equation

dy dx +^ P^ (x)y^ =^ Q(x)

Examples: dy dx

• x^2 y = x^3

dy dx + 2y^ =^ e

x

dy dx =^

x − y x + y (x + y)dy = (x − y)dx (y − x)dx + (x + y)dy = 0

This is now in the form:

M dx + N dy = 0

dy dx

1 − y^2 2 xy − sin y (2xy − sin y)dy = (1 − y^2 )dx

(y^2 − 1)dx + (2xy − sin y)dy = 0

Review from Calculus III: If z is a function of x and y then:

dz =

∂z ∂x dx^ +^

∂z ∂y dy