# Differential Equations Assignment (small step needs clarification)

• Mozart
In summary: Otherwise, you would write it as [(λ - alpha)(λ - beta)(λ - gamma)][e^(λx)]. In summary, when solving differential equations, you can use a characteristic equation to find the roots and then make a substitution to simplify the equation. When the roots are repeated, you can write them as [(λ - alpha)^n][e^(λx)].
Mozart
When given

y'' + ay' + by + c = 0

The characteristic equation is:

λ^2 + aλ + b = 0

Now I'm making a substitution of e^(λx) to get

[(λ^2 + aλ + b)][e^(λx)] = 0

double root : [(λ - alpha)^2] [e^(λx)] = 0

My question now is that if I were to do the same thing with the following equation would it follow that

y''' + ay'' + by' + cy +d = 0

Characteristic equation λ^3 + aλ^2 + bλ + c

then making the same substitution ( λ^3 + aλ^2 + bλ + c ) e^(λx)

NOW HERE IS MY QUESTION: Can I write the above as

[(λ - alpha)^3][e^(λx)]

or do I have to write it a different way. Thank you very much.. I know this doesn't really have much to do with the solving of differential equations but it would be very helpful if I knew that that was correct for my assignment.

Thanks.

Mozart said:
NOW HERE IS MY QUESTION: Can I write the above as

[(λ - alpha)^3][e^(λx)]

or do I have to write it a different way. Thank you very much.. I know this doesn't really have much to do with the solving of differential equations but it would be very helpful if I knew that that was correct for my assignment.

Thanks.

You can write it like that, if the roots are repeated.

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to describe how a quantity changes over time or space.

## 2. What is the difference between an ordinary and a partial differential equation?

An ordinary differential equation involves a single independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations describe the behavior of a single variable, while partial differential equations describe the behavior of a system with multiple variables.

## 3. How do you solve a differential equation?

The method for solving a differential equation depends on its type and complexity. Some methods include separation of variables, substitution, and using differential equation solvers such as Euler's method or Runge-Kutta methods. It is important to first identify the type of differential equation and the initial conditions before choosing a method.

## 4. What are some real-world applications of differential equations?

Differential equations are used in a wide range of scientific fields, including physics, engineering, economics, and biology. They are used to model and understand complex systems such as population growth, fluid mechanics, electrical circuits, and chemical reactions.

## 5. How can I check if my solution to a differential equation is correct?

You can check your solution by plugging it back into the original differential equation and seeing if it satisfies the equation. You can also check for any boundary or initial conditions that need to be met. Additionally, you can compare your solution to known solutions or use a differential equation solver to verify the correctness of your solution.

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