Differential Equations
by Karolina Drewnik
A differential equation
is a mathematical equation that relates some function
of one or more variables
with its derivatives.
- Mechanical Systems
- Electrical Circuits
- Population Models
- Newton's Law of Cooling
- Compartmental Analysis
Examples of First-Order Differential Equations
general form
first-order ordinary differential equation
t - independent variable
y(t) - dependent variable
The goal: determine the unknown function y(t) whose derivative satisfies the above condition and which passes through the point
qualitative analysis
of First-Order Ordinary Differential Equations
AUTONOMOUS
NON-AUTONOMOUS
example:
example:
QUALITATIVE ANALYSIS
OF THE AUTONOMOUS ODE
QUALITATIVE ANALYSIS
OF THE NON-AUTONOMOUS ODE
Directly Integrable
First-Order Differential Equations
general solution:
Separable
First-Order Differential Equations
Solution:
Equation of type
Solution:
First step:
Second step:
SEPARABLE METHOD
(to compute y')
Equation of type
Solution:
First step:
Second step:
SEPARABLE METHOD
(to compute y')
Linear
First-Order Differential Equations
Solution:
First step:
Second step:
SEPARABLE METHOD
Bernoulli
First-Order Differential Equations
Solution:
First step:
Second step:
LINEAR METHOD
to obtain
- Mechanical Systems
- Electrical Circuits
- One-Dimensional Free-Fall Motion
Examples of Second-Order Differential Equations
general form
second-order ordinary differential equation
t - independent variable
y(t) - dependent variable
The goal: determine the unknown functions that satisfies the above ordinary differential equation
on some interval
INITIAL CONDITIONS:
Reduction of Order for Problems
of the form
Solution:
Convert
into first-order differential equation
Reduction of Order for Problems
of the form
Solution:
Write
Linear
Second-Order Differential Equations
Solution:
HOMOGENOUS
NONHOMOGENOUS otherwise
HOMOGENOUS
NONHOMOGENOUS
Constant-Coefficient Homogeneous ODE
Solution:
Uniform solution:
The Variation of Parameters Formula
Solution:
1. Find solution like in previous method
2. Use the variation of parameters - change parameters to C1, C2
3. Create system of equations
Laplace transform
Solution:
Thank you
Differential Equations
By lubiewarzywa
