first order differential equations

a differential equation

equation connecting any of x, y , dy/dx , d^2y/dx^2

order of a differential equation

highest power of the derivative that occurs

first order differential equation

involves nothing more complex than dy/dx

solution to a differential equation

equation linking x and y

y=x^2 +x + c

general solution of the DE - describes all curbed whose equations satisfy the DE and involves arbitrary constant c

x^2 +x+3

particular solution of the DE

When dy/dx = f(x)g(y) (it is given in terms of two different variables ) then we can

rearrange and write
/ 1/g(y) dy = /f(x) dx
this is called seperation of the variables

we can only separate the variables by

multiplying /
dividing both sides (not adding or subtracting)
eg. dy/dx=x^2y > /1/y dy = /x^2 dx
but for dy/dx =x^2 +y we cannot seperate the variables

when formulating differential equation problems you will often be given a statement

linking a rate of change and a quantity to which it is proportional to in some way

x increases at a constant rate

dx/dt = k

x decreases at a constant rate

dx/dt = -k

x increases at a rate which is proportional to x . this is exponential growth

dx/dt =kx

x decreases at a rate which is proportional to x . this is exponential decay

dx/dt= -kx

x increases at a rate which is inversely proportional to x

dx/dt = k/x

x decreases at a rate which is proportional to x^3

dx/dt = -kx^3

x increases at a rate which is proportional to A- root x

dx/dt = k(A-root x)

for some DE questions you may need
to use related rates of change (A2 diff 3) to set up the de

the key is ensure the variables in the DE match up with the variables in the rate statement. A DE involving da/dt must be written exclusively in terms of
a and t with no other variables present