2. LS
Objectives
To define and solve a
simple separable
differential equation of
first order.
1
To Identify a differential
equation and its order.
MATH
2
4. Activity 1
Given the function f defined over
with
1) Find .
2) Show that satisfy the equation .
3) Verify that the functions g and h defined over with:
and are also solutions of (E).
5. Definition 1
A differential equation is every equation relating a function y
of an independent variable x to
its first derivative, its successive higher order derivatives and
the real variable x.
Notation:
If the function is denoted by then its successive derivatives
denoted by:
(1st
derivative), (2nd
derivative), …
6. Definition 2
• The order of a D.E. is the order of the highest order
derivative of y that appears in the equation.
• Examples:
First order
First order
Second order
7. Definition 3
A differential equation of first order is a relation between
an unknown function defined on an interval I, its
derivative and the variable x.
Examples:
, such that is defined on .
, such that is defined on
8. Definition 4:
A solution of a differential equation on an interval I
is any function defined and differentiable over I that
satisfies the given equation for every x in I .
Example:
The functions
and
are solutions of
9. Activity 2
Given the function defined on such that
.
1) Find
2) Find y such that .
,
So
¿ 𝑥2
− 𝑥 +𝑐 /𝑐 ∈ 𝐼𝑅
(1)2
− (1)+𝑐=3
,
10. • Property 1
Let f be a continuous function over the interval I.
The solutions of the differential equation
are given by , where c is any constant real number .
11. Activity 3
Given the differential equation (E):
1) Find .(the general solution of this equation).
then
So:
𝑦=√2(𝑥+𝑐) or 𝑦=−√2(𝑥+𝑐)
Implicit General Solution
Explicit General Solutions
12. 2) Find a particular solution y, of this equation such that and
verifies the condition
Knowing that the explicit general solutions are:
but
then: and
A particular explicit solution of (E) is
𝑦 (0)=1
√2(0+𝑐)=1
, 𝑐=
1
2
𝑦=√2(𝑥+𝑐) or 𝑦=−√2(𝑥+𝑐)
13. Definition 5
Separable equations
A separable first order differential equation is a differential
equation of the form with and are two real functions.
The solutions of a separable equation are given by:
, where c is a real constant
14. Application
Given the differential equation (E) , where y is a non-zero real
function.
1. Find the general solution of (E).
, ,
then ,
thus.
Implicit General Solution
Explicit General Solutions
15. 2) Find a particular solution of (E) that verifies the condition
is the general solution of (E).
then
thus a particular solution of (E) is
16. 3) Let g be a particular solution of (E) and denote by (C) its
representative curve in an orthonormal system of axes. Find g(x)
such that the tangent to (C) at its point A of abscissa 0 is parallel to
the line .
and
Then and
so ,
Hence
