First-Order Equations: Separable, Homogeneous and Linear

Name: First-Order Equations: Separable, Homogeneous and Linear
Author: StemPy Lab

Most equations of this unit have a first derivative and nothing higher. Three methods solve nearly all of them. Separate the variables when the two sides can be pulled apart, put $y = vx$ when the equation is homogeneous, and multiply by an integrating factor when the equation is linear. Learn to choose the method from the shape of the equation. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Variables separable
A first-order equation is separable when it can be written as $\dfrac{dy}{dx} = f(x)\,g(y)$ . Then all the $y$ terms go to one side and all the $x$ terms to the other, and each side is integrated on its own. The division by $g(y)$ can lose constant solutions, so we list those separately.
1 simulation1 practice question1 Python program4 problems1 Python lab
Read the lesson
Lesson 2about 25 min
Homogeneous equations
A homogeneous equation has a slope that depends only on the ratio $\dfrac{y}{x}$ , so $\dfrac{dy}{dx} = F\!\left(\dfrac{y}{x}\right)$ . The substitution $y = vx$ turns it into a separable equation in $v$ and $x$ . After integrating, we put $v = \dfrac{y}{x}$ back. When $x$ is the simpler unknown, we use $x = vy$ .
1 simulation1 practice question1 Python program3 problems
Read the lesson
Lesson 3about 25 min
Linear equations and the integrating factor
A linear first-order equation has the form $\dfrac{dy}{dx} + Py = Q$ , with $P$ and $Q$ functions of $x$ alone. Multiplying by the integrating factor $e^{\int P\,dx}$ turns the left side into the derivative of a product. Then one integration finishes the job. The solution is a transient part plus a steady part.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
Read the lesson

Test Your Learning

First-Order Equations: Timed Test

10 questions · 25 min

Not started

Your teacher

StemPy Lab

← Differential Equations

← Differential Equations: Order, Degree and Solutions Modelling with Differential Equations →

More onFunctions and Calculus › Differential Equations

Discussion (0)

Loading…

First-Order Equations: Separable, Homogeneous and Linear

Course outline (3 lessons)

Variables separable

Homogeneous equations

Linear equations and the integrating factor

Test Your Learning

Your teacher

1 follower

Discussion (0)

Want to go further?