Differential Equations

Now we will add income to our model in addition to consumption and returns on savings. In this case,
the net change in our wealth can be written as:
x!(t) = r(t)x(t)+ y(t)− c(t) (8.12)
Here r is the rate of return on our wealth, y(t) is our income at time t, c(t) is our consumption at time t.
Notice that this equation is quite general.

Now we will add income to our model in addition to consumption and returns on savings. In this case,
the net change in our wealth can be written as:
x!(t) = r(t)x(t)+ y(t)− c(t) (8.12)
Here r is the rate of return on our wealth, y(t) is our income at time t, c(t) is our consumption at time t.

Now we will add income to our model in addition to consumption and returns on savings. In this case,
the net change in our wealth can be written as:
x!(t) = r(t)x(t)+ y(t)− c(t) (8.12)
Here r is the rate of return on our wealth, y(t) is our income at time t, c(t) is our consumption at time t.

If y(t)=c(t) and r(t)=0 at all times, t, then we end up with the safe box example

If y(t)=0, r(t)=0 and c(t)=0.03x(t), we end up with the cake eating example.

If y(t)=c(t) and r(t)=0.03, we end up with the returns on wealth example.

x^’= ax + c (8.18)

This is a first-order linear ordinary differential equation with constant coefficients

There is a single
variable involved, so it is a scalar system.

A single variable is involved, so it is a scalar system.