Differential equations are mathematical expressions involving an unknown function and its derivatives, essential for describing many physical phenomena. Ordinary differential equations (ODEs) focus on functions of a single independent variable and their derivatives. A notable type of ODE is the sep...
Overview of Differential Equations
Kosala Madhushanka
september 2024
1 Definition and Types of Differential Equations
A differential equation is an equation that involves an unknown function and
its derivatives. Ordinary differential equations (ODEs) involve functions of one
independent variable and their derivatives.
2 Separable Differential Equations
2.1 Definition
A first-order ODE is separable if it can be written in the form
dy f (x)
= .
dx g(y)
2.2 Solution Method
• Separate the variables:
dy
= f (x)dx.
g(y)
• Integrate both sides: Z Z
dy
= f (x)dx.
g(y)
3 Initial Conditions
3.1 Definition
An initial condition specifies the value of the solution at a particular point.
3.2 Initial Value Problem
A differential equation together with an initial condition is known as an initial
value problem.
1
, 3.3 Solving Initial Value Problems
1. Solve the differential equation.
2. Apply the initial condition to find the particular solution.
4 Exponential Growth and Decay Models
The separable differential equation for exponential growth and decay is
dy
= k · y.
dx
The solution is
y(x) = c · ekx ,
where c is a constant determined by the initial condition.
5 Homogeneous and Non-Homogeneous Differ-
ential Equations
5.1 Homogeneous Differential Equation
A differential equation is homogeneous if it equals zero when both sides are
divided by the highest order derivative.
5.2 Non-Homogeneous Differential Equation
A differential equation is non-homogeneous if it is not homogeneous.
6 Identifying and Solving First-Order Differen-
tial Equations
1. Identify the type of differential equation.
2. Apply the appropriate solution method.
3. If the differential equation is separable, solve it using separation of vari-
ables.
7 Undetermined Coefficient Method for Non-
Homogeneous Equations
A method for finding particular solutions to non-homogeneous equations in-
volves guessing a form for the particular solution and solving for the coefficients.
2
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller erandagunawardana58. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $7.49. You're not tied to anything after your purchase.
