Homogeneous and Heterogeneous Calculator

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Example

Created on 2024-06-20Asked by Aiden Green (Solvelet student)

Classify the differential equation

y'' + 3y' + 2y = e^x

as homogeneous or heterogeneous.

Solution

To classify the differential equation

y'' + 3y' + 2y = e^x

: A differential equation is homogeneous if it can be written in the form:

L(y) = 0

Where

L

is a linear differential operator. The given differential equation is:

y'' + 3y' + 2y = e^x

Since the right-hand side

e^x \neq 0

, the differential equation is not homogeneous. It has a non-zero forcing term. Therefore, the differential equation

y'' + 3y' + 2y = e^x

is heterogeneous. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Mateo Martin on Solvelet

1. Determine whether the system of equations

\{2x + 3y = 0, 4x - 6y = 0\}

is homogeneous or heterogeneous,2. Find the particular solution to the homogeneous differential equation

\frac{dy}{dx} = -\frac{2x}{y}

by separation of variables and verify its correctness.,

DefinitionIn mathematics and particularly in physics, a homogeneous space is a space (i.e. a set with some structure) that is in some sense the same at each point. Heterogeneous systems (equations that have non-uniform properties or that contain terms of different degrees on the left hand side of an equation. Homogeneous systems are easier to analyze and homogenize while methods for heterogeneous systems are often more complex. Example: Differential equation y′′+3y′+2y=0 is homogeneous, however y′′+3y′+2y=ex is non-homogeneous because the right hand side is unequal to zero.

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