First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
Linear
Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
Constant Coefficients
Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
Cauchy-Euler Equations
Chebyshev Equations
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Classify Differential Equations
Fourier Series
Slope Fields
Wronskian
Existence and Uniqueness
Boundary Value Problems
Euler's Method
Inhomogeneous ODE's
Resonance
Partial Differential Equations
Linear Systems
Exponential Growth/Decay
Population Dynamics
Projectile Motion
Chemical Concentration
Fluids (Mixing)
Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > first-order, linear

### Differential Equations Alpha List

 Boundary Value Problems Cauchy-Euler Equations Chebyshev Equations Chemical Concentration Classify Differential Equations Constant Coefficients Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay First Order, Linear Fluids (Mixing) Fourier Series Inhomogeneous ODE's Integrating Factors (Exact) Integrating Factors (Linear) Laplace Transforms Linear Systems Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance Second Order, Linear Separation of Variables Shifting Theorems Slope Fields Solve Initial Value Problems Square Wave Substitution Undetermined Coefficients Unit Impulse Function Unit Step Function Variation of Parameters Wronskian

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First-Order, Linear Differential Equations

There are several techniques for solving first-order, linear differential equations. We list them here with links to other pages that discuss those techniques. But before you leave, let's discuss what a first-order, linear ODE is and look at some easier techniques that will save you some time and energy.

Type

Form

Technique

Homogeneous

$$y' = f(y/x)$$

Substitution

Basic First-Order, Linear

$$y' + p(t)y = g(t)$$

Integrating Factor

 What Is A First-Order, Linear Differential Equation?

A first-order equation is one in which the highest derivative is a first derivative. A linear equation is a bit more complicated to determine. This short video clip explains this and shows some examples.

 Dr Chris Tisdell - linear equation
 How To Solve

Okay, now that you know what a first-order, linear ODE is, we can tell you that there are several techniques to solve them, some of which are more rigorous than others. We listed a few at the top of the page but, before you jump there, take a minute to look at your differential equation. It may already be in a form that can be evaluated directly. If you can learn to recognize one of these forms, it may save you some time and energy on your homework or exam. Also, it will help you to start to get a feel for what to look for.

Sometimes a differential equation may already be in either derivative product rule or quotient rule form. This video clip shows some examples and explains what to look for.

 Dr Chris Tisdell - examples

Here are a couple of videos containing in-depth discussions of first-order, linear differential equations with constant coefficients. Given an equation of the form $$y' + ky = g(t)$$, the solution is $$\displaystyle{ y = e^{-kt}\int{e^{kt}g(t)~dt} }$$. This is easily found from the use of integrating factors. The videos have discussion relating to steady-state and transient solutions and some specific examples of functions for $$g(t)$$. They are good to watch to get a feel for these types of differential equations.

 MIT OCW - in-depth discussion
 How To Solve Any First-Order Differential Equation

This is a good video to watch to get an idea of what is coming up in your study of first-order equations. It puts together the techniques in an entertaining and informative way. However, you will not understand most of it until you have studied more differential equations techniques. Once you have more experience with first-order differential equations, come back here and watch this video again.

 Krista King Math - various techniques
 Okay, so you are now ready to start working with first-order, linear differential equations. We suggest one of the topics below. next: integrating factors → next: exact equations → next: substitution →

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