Differential Equations
Rates of Change
dy
● dx represents the change in y with respect to x
dy
● When time is used instead of x then dt represents the rate of change of y
● Examples:
dt = rate of change of velocity (or acceleration)
○ dv
○ dx
dt = rate of change of displacement (or velocity)
○ If m is mass then dm
dt = rate of change of mass
Formulating Differential Equations
● You can use differential equations to model the rate of change of a
variable
● If a is proportional to b then a = k b , and we can find k (the constant of
proportionality) by plugging in known values of a and b
● We can use this idea to formulate a differential equation:
○ If the rate of change of temperature of a glass of water is
proportional to its volume then dTdt = kV
○ If the acceleration of an object is inversely proportional to its mass
squared then dv k
dt = m2
Solving Differential Equations
dy
● Solve differential equations to get equation in terms of y instead of dt
● Can solve through indirect integration:
○ If not enough information to find c then the equation you have is the
general solution
○ If you can sub in values to find c then the new equation (after
subbing to find c ) is a particular solution
● Can solve through separating variables:
dy
○ Equation like dt = 3y 2 is in terms of y , not t , so we need to integrate
with respect to t
1 dy
○ y 2 dt = 3
dy
○ ∫ y1 2 dt dt = ∫ 3 dt
○ ∫ y1 dy 2 = 3t
1
○ − y = 3t + c
dy
○ As a general rule: dt = k f (y) ⇒ f ′ (y) = kt + c
Rates of Change
dy
● dx represents the change in y with respect to x
dy
● When time is used instead of x then dt represents the rate of change of y
● Examples:
dt = rate of change of velocity (or acceleration)
○ dv
○ dx
dt = rate of change of displacement (or velocity)
○ If m is mass then dm
dt = rate of change of mass
Formulating Differential Equations
● You can use differential equations to model the rate of change of a
variable
● If a is proportional to b then a = k b , and we can find k (the constant of
proportionality) by plugging in known values of a and b
● We can use this idea to formulate a differential equation:
○ If the rate of change of temperature of a glass of water is
proportional to its volume then dTdt = kV
○ If the acceleration of an object is inversely proportional to its mass
squared then dv k
dt = m2
Solving Differential Equations
dy
● Solve differential equations to get equation in terms of y instead of dt
● Can solve through indirect integration:
○ If not enough information to find c then the equation you have is the
general solution
○ If you can sub in values to find c then the new equation (after
subbing to find c ) is a particular solution
● Can solve through separating variables:
dy
○ Equation like dt = 3y 2 is in terms of y , not t , so we need to integrate
with respect to t
1 dy
○ y 2 dt = 3
dy
○ ∫ y1 2 dt dt = ∫ 3 dt
○ ∫ y1 dy 2 = 3t
1
○ − y = 3t + c
dy
○ As a general rule: dt = k f (y) ⇒ f ′ (y) = kt + c
