Description

Differential Equations Notes

Categories

Published

All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Share

Transcript

1
Math 230 – Section 1.1 Basic definitions and terminology I.
Differential equations
Definition: A
differential equation
(DE) is an equation containing the derivatives or differentials of one or more dependent variables, with respect to one or more independent variables. Ex: Newton’s equation
2
2
=
(
,
)
Classification
Type:
: by type, order, and linearity 1.
Ordinary differential equation (ODE): contains only
ordinary
derivatives of one or more dependent variables with respect to a
single
independent variable. Ex : 2.
Partial differential equation (PDE): contains
partial
derivatives of one or more dependent variables with respect to
2 or more
independent variables. Exa)
Laplace equation : b)
Heat equation c)
Wave equation
2
Order:
The order of a DE is the highest derivative’s order that appears in the equation. Ex : The general form for the n-th order ODE is
�
,
,
,
2
2
,
⋯
,
=0
Linearity:
1.
Linear: Ex : The general form for a linear ODE is
(
)
+
−
1
(
)
−
1
−
1
+
⋯
+
2
(
)
2
2
+
1
(
)
+
0
(
)
=
(
)
2.
Nonlinear: Ex :
3
II.
Solution of a differential equation
Definition: A function
defined on an interval
is said to be a solution of a DE if it reduces the equation to an identity when substituted into the equation. Ex : verify that the indicated function is a solution of the given differential equation.
a)
+20
=24;
=65
−
65
−
20
b)
′′
+
=sec
;
=cos
ln(cos
)+
sin
for 0<
<
2
4
c) 2
+(
2
+2
)
=0;
2
+
2
=
where
is a constant.
Definitions1.
Trivial solution: : 2.
Explicit solution: 3.
Implicit solution:
n
-parameter family of solutions: ExConsider the ODE
′
=
1/2
: one-parameter family of solutions

Search

Similar documents

Tags

Related Search

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks

SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...Sign Now!

We are very appreciated for your Prompt Action!

x