differential equations mathrmatics

1. Lesson 2: Basics of
Differential Equation

2. Calculus
►is the mathematics of change, and rates of
change are expressed by derivatives. Thus,
one of the most common ways to use
calculus is to set up an equation containing
an unknown function y=f(x) and its
derivative, known as a differential
equation. Solving such equations often
provides information about how quantities
change and frequently provides insight into
how and why the changes occur.

3. General Differential Equations
►differential equation - is an equation
involving an unknown function y=f(x) and
one or more of its derivatives.
► Solution to a differential equation- is a
function y=f(x) that satisfies the
differential equation when f(x) and its
derivatives are substituted into the equation.

4. Example of Differential Equations
Equation Solution
y′=2x y=x2
y′+3y=6x+11 y=e−3x+2x+3
y′′−3y′+2y=24e−2x
y=3ex−4e2x+2e−2x
Note that a solution to a differential equation is not necessarily unique, primarily because the
derivative of a constant is zero. For example, y=x2+4 is also a solution to the first differential
equation

5. Verifying Solutions of Differential Equations
Example:
►Verify that the function y=e−3x+2x+3 is a
solution to the differential equation
y'+3y=6x+11 .

6. solution
► Given: y=e−3x+2x+3
then y'=−3e−3x+2 Note: f(x) = ekx then f´(x) = kekx
Substitute: the value of y and y’ to the equation: y'+3y=6x+11
(−3e−3x+2) + 3 (e−3x+2x+3) = 6X + 11
−3e−3x+2+3e−3x+6x+9 = 6X +11
6X+11 =6X +11

7. Verify that the equation y=2e3x−2x −2
is a solution to the differential
equation y'−3y=6x+4.
Y = 2e3x-2x-2
Y’ = 6e3x-2
Substitute:
(6e3x-2) – 3(2e3x-2x-2) = 6x +4
6e3x – 2 – 6e3x + 6x + 6 = 6x +4
6x +4 = 6x+ 4

8. Definition: order and degree of a
differential equation
► The order of a differential equation is the highest
order of any derivative of the unknown function
that appears in the equation
► The degree of a differential equation is the power
of the highest order derivative of the equation.

9. Identifying the Order and degree of a
Differential Equation

10. Answer

12. General and Particular Solutions
When we first performed integrations, we obtained
a general solution (involving a constant, K).
We obtained a particular solution by substituting
known values for x and y. These known conditions are
called boundary conditions (or initial conditions).
It is the same concept when solving differential
equations - find general solution first, then substitute
given numbers to find particular solutions.

13. Example: