# Heat equation solving quadratic equation with complex numbers

• dp182
In order to get the equation in the form they want, you need to find values for γ and δ that will give you the ± sign, which will then give you the imaginary part for β.
dp182

## Homework Statement

given that kλ2-ρcpuλ-ρcpωi=0
plug into the quadratic formula and get out an equation that looks like this

λ=α+iβ±γ√(1+iδ) where α,β,γ,and δ are in terms of ρ,cp,u,k, and ω

(-b±√b2-4ac)/2a
2-ρcpuλ-ρcpωi=0
λ=α+iβ±γ√(1+iδ)

## The Attempt at a Solution

so I plugged it in and came out with
(ρcpu/2k)±(ρcpu/2k)√1+(4kωi/ρcpu2)
so γ=(ρcpu/2k) and δ=(4kω/ρcpu2)
but I'm unable to make the first term α+iβ and help would be great

dp182 said:

## Homework Statement

given that kλ2-ρcpuλ-ρcpωi=0
plug into the quadratic formula and get out an equation that looks like this

λ=α+iβ±γ√(1+iδ) where α,β,γ,and δ are in terms of ρ,cp,u,k, and ω

(-b±√b2-4ac)/2a
2-ρcpuλ-ρcpωi=0
λ=α+iβ±γ√(1+iδ)

## The Attempt at a Solution

so I plugged it in and came out with
(ρcpu/2k)±(ρcpu/2k)√1+(4kωi/ρcpu2)
Should be λ= (ρcpu/2k)±(ρcpu/2k)√1+(4kωi/ρcpu2) since you are solving the quadratic equation for λ. Note that I didn't check your work.

dp182 said:
so γ=(ρcpu/2k) and δ=(4kω/ρcpu2)
but I'm unable to make the first term α+iβ and help would be great

They're not asking you to make the first term α+iβ, just that you have an equation that has the form they give. In what you have, α = ρcpu/(2k) and β = 0.

## 1. What is the heat equation?

The heat equation is a partial differential equation that describes the distribution of heat (or temperature) in a given space over time. It is commonly used in physics, engineering, and other scientific fields to model heat transfer and thermal diffusion.

## 2. How is the heat equation related to solving quadratic equations with complex numbers?

The heat equation can be solved using the method of separation of variables, which involves breaking down the equation into simpler parts that can be solved individually. This process results in a quadratic equation with complex coefficients, which can be solved using standard techniques for solving quadratic equations.

## 3. What are complex numbers and how are they used in solving the heat equation?

Complex numbers are numbers that have both a real and an imaginary component, and are represented as a + bi, where a and b are real numbers and i is the imaginary unit (√-1). In solving the heat equation, complex numbers are used to represent the oscillatory behavior of the solution, which is a key characteristic in the distribution of heat.

## 4. What are the benefits of using complex numbers in solving the heat equation?

Using complex numbers allows for a more complete and accurate representation of the solution to the heat equation, as it takes into account both the real and imaginary components of the temperature distribution. It also allows for easier and more efficient calculations, as complex numbers have convenient mathematical properties.

## 5. Are there any limitations to using complex numbers in solving the heat equation?

One potential limitation is that the use of complex numbers may not accurately model certain physical systems that do not exhibit oscillatory behavior, as the heat equation assumes. Additionally, the use of complex numbers may add complexity to the calculations and may be more difficult to interpret compared to solutions using only real numbers.

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