ordinary differential equation

An ordinary differential equation is a differential equation that depends on only one variable. It can be represented with a phase space. The implicit differential equation takes the form:

• is the -th derivative of wrt

Example:

1. Gravitational acceleration
2. Speed:

It requires only one initial condition to find

Strategy

Solve a Separable Differential Equation

1. Separate variables by rewriting the differential equation in the format, where only depends on , and only depends on

2. Move terms with the same variables on one side:

3. Integrate both sides and solve for

4. (optional) Use initial conditions to find
5. Plug and back into the differential equation (pick any value for integration constant if there’s none)
6. (optional) Plot slope field and the solution to see if they match

# Sage
y = function('y')(x)
de = diff(y,x) + y-2
h = desolve(de, y)
h2 = desolve(de, y, ics=[0, 3]) # with initial conditions
solve(h2, y)

Geographical

If the ordinary differential equation is not separable, use

slope field

a graphical representation of the solutions to a ordinary differential equation

Plug different set of values for into , and sketch the slope

# Sage
var('x,y')
plot_slope_field(x+y, (x, -4, 4), (y, -4, 4))

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