First Order ODEs – Comprehensive Study Guide
FIRST ORDER
ORDINARY DIFFERENTIAL
EQUATIONS
A Comprehensive Study Guide
Compiled from Three Standard Textbooks:
Zill – A First Course in Differential Equations (12th ed.)
Boyce, DiPrima & Meade – Elementary Differential Equations (12th ed.)
Nagle, Saff & Snider – Fundamentals of Differential Equations (9th ed.)
StudyPool Educational Series
2026
Page 1
, First Order ODEs – Comprehensive Study Guide
TABLE OF CONTENTS
1. Introduction and Fundamental Concepts....................................................................................3
2. Direction Fields and Autonomous Equations.............................................................................5
3. Separable Equations....................................................................................................................7
4. Linear First Order Equations...................................................................................................... 9
5. Exact Equations........................................................................................................................ 12
6. Integrating Factors for Non-Exact Equations........................................................................... 14
7. Substitution Methods: Homogeneous and Bernoulli Equations...............................................16
8. Existence and Uniqueness Theorem.........................................................................................19
9. Engineering Applications and Mathematical Modeling...........................................................20
10. Numerical Methods: Euler and Runge-Kutta.........................................................................23
References................................................................................................................................26
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, First Order ODEs – Comprehensive Study Guide
1. Introduction and Fundamental Concepts
A differential equation (DE) is an equation that contains the derivatives of one or more
unknown functions with respect to one or more independent variables. When the equation involves
only ordinary derivatives with respect to a single independent variable, it is called an ordinary
differential equation (ODE). When partial derivatives appear, it is called a partial differential
equation (PDE). This guide focuses exclusively on first order ODEs.
1.1 Order and Linearity
The order of a differential equation is the order of the highest derivative present. A first order
ODE involves only dy/dx (or y′) and no higher derivatives. The general form of a first order ODE
is:
or equivalently in normal form:
A first order ODE is linear if it can be written in the form:
The dependent variable y and its derivative dy/dx both appear to the first power, are not
multiplied together, and do not appear inside nonlinear functions such as sin(y) or . Any first order
ODE that does not satisfy these conditions is nonlinear.
1.2 Solutions and Initial Value Problems
A function y = φ(x) is a solution of a differential equation on an interval I if, when substituted
into the equation, it reduces it to an identity for all x in I. An initial value problem (IVP) consists
of a differential equation together with an initial condition y() = . The initial condition determines a
unique solution from the family of all solutions.
Definition: Initial Value Problem (IVP)
Find a function y = φ(x) such that:
The solution is a function that satisfies both the equation and the prescribed value at x = .
1.3 Classification of First Order Methods
Method Applicable Form Key Idea
Separation of Variables dy/dx = g(x)h(y) Separate x and y on different sides,
Page 3
FIRST ORDER
ORDINARY DIFFERENTIAL
EQUATIONS
A Comprehensive Study Guide
Compiled from Three Standard Textbooks:
Zill – A First Course in Differential Equations (12th ed.)
Boyce, DiPrima & Meade – Elementary Differential Equations (12th ed.)
Nagle, Saff & Snider – Fundamentals of Differential Equations (9th ed.)
StudyPool Educational Series
2026
Page 1
, First Order ODEs – Comprehensive Study Guide
TABLE OF CONTENTS
1. Introduction and Fundamental Concepts....................................................................................3
2. Direction Fields and Autonomous Equations.............................................................................5
3. Separable Equations....................................................................................................................7
4. Linear First Order Equations...................................................................................................... 9
5. Exact Equations........................................................................................................................ 12
6. Integrating Factors for Non-Exact Equations........................................................................... 14
7. Substitution Methods: Homogeneous and Bernoulli Equations...............................................16
8. Existence and Uniqueness Theorem.........................................................................................19
9. Engineering Applications and Mathematical Modeling...........................................................20
10. Numerical Methods: Euler and Runge-Kutta.........................................................................23
References................................................................................................................................26
Page 2
, First Order ODEs – Comprehensive Study Guide
1. Introduction and Fundamental Concepts
A differential equation (DE) is an equation that contains the derivatives of one or more
unknown functions with respect to one or more independent variables. When the equation involves
only ordinary derivatives with respect to a single independent variable, it is called an ordinary
differential equation (ODE). When partial derivatives appear, it is called a partial differential
equation (PDE). This guide focuses exclusively on first order ODEs.
1.1 Order and Linearity
The order of a differential equation is the order of the highest derivative present. A first order
ODE involves only dy/dx (or y′) and no higher derivatives. The general form of a first order ODE
is:
or equivalently in normal form:
A first order ODE is linear if it can be written in the form:
The dependent variable y and its derivative dy/dx both appear to the first power, are not
multiplied together, and do not appear inside nonlinear functions such as sin(y) or . Any first order
ODE that does not satisfy these conditions is nonlinear.
1.2 Solutions and Initial Value Problems
A function y = φ(x) is a solution of a differential equation on an interval I if, when substituted
into the equation, it reduces it to an identity for all x in I. An initial value problem (IVP) consists
of a differential equation together with an initial condition y() = . The initial condition determines a
unique solution from the family of all solutions.
Definition: Initial Value Problem (IVP)
Find a function y = φ(x) such that:
The solution is a function that satisfies both the equation and the prescribed value at x = .
1.3 Classification of First Order Methods
Method Applicable Form Key Idea
Separation of Variables dy/dx = g(x)h(y) Separate x and y on different sides,
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