# Differential Equations

Using the GeoGebra command solveODE you can illustrate numerical solutions to first and second order ordinary differential equations.

## First order differential equations

When using the command solveODE, $$y$$ must be a function of $$x$$. Let

$\frac{dy}{dx} = f(x,y)$

then you can used the commands:

solveODE(f)
solveODE(f, <Point on f>)
solveODE(f, <Start x>, <Start y>, <End x>, <Step>)

and:

SlopeField(f)
SlopeField(f, <Number n>)
SlopeField(f, <Number n>, <Length multiplier a>)

If the solution curve has vertical points and if the equation can be written as

$\frac{dy}{dx} = \frac{f(x,y)}{g(x,y)}$

it is better to use the command

solveODE(f, g, x(A), y(A), <End t>, <Step> )

where A is a point on the curve. This command is used in the worksheet above.

## Second order differential equations

The command

SolveODE[<b(x)>,<c(x)>,<f(x)>,<Start x>,<Start y>,<Start y'>,<End x>,<Step> ]

solves the equation

$y''+b(x)y'+c(x)y=f(x)$

## Exercises

#### Exercise 1

First order ODE

Make the function

$f(x, y) = y\sin(x) + \frac{y}{x}.$

Link an input box to the function so you easily can redefine it.

Make a point $$A$$.

Make a slope field and solve the equation numerically.