You need to have the AutoPostBack property set to true for RadioButtonList1
edited:
And yes you will need to leave the AutoPostBack in the mark up
protected void RadioButtonList1_SelectedIndexChanged(object sender, EventArgs e)
{
if (RadioButtonList1.SelectedValue == "1")
{
equationPanelExtender.Collapsed = false;
variablePanelExtender.Collapsed = true;
}
if (RadioButtonList1.SelectedValue == "2")
{
equationPanelExtender.Collapsed = true;
variablePanelExtender.Collapsed = false;
}
}