Finding the maximum size of an Initial Value Problem coefficient

In summary, the given conversation discusses finding the maximum size of the coefficient v in the IVP equation for temperature T in a tub filled with water. The equation is diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39. The problem is to calculate the maximum value of v for a person taking a 40-minute bath, with the condition that the water in the tub should not fall below 37 degrees Celsius. The screenshot shows a solution using Maple's dsolve function, but the method for finding the maximum value of v is unclear.
  • #1
MathMan2022
12
1
Homework Statement
find the maximum size of a IVP coefficient size?
Relevant Equations
diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39
The following IVP
diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39

Describes the tempetatur T in celcius at the time x of a tub filled with water. A tub which is filled with hot water at rate of v l/min.

Lets say I am told that a guy takes a 40 min bath, and during those 40 min he doesn't want the water in the tub to become colder than 37 c.
How would I go about calculating the maximum size of v for that periode?
 
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  • #2
MathMan2022 said:
Homework Statement:: find the maximum size of a IVP coefficient size?
Relevant Equations:: diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39

The following IVP
diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39

Describes the tempetatur T in celcius at the time x of a tub filled with water. A tub which is filled with hot water at rate of v l/min.

Lets say I am told that a guy takes a 40 min bath, and during those 40 min he doesn't want the water in the tub to become colder than 37 c.
How would I go about calculating the maximum size of v for that periode?

Welcome to PF.

Can you show us your work so far? We are not allowed to help you until we see your initial work. Thanks.
 
  • #3
berkeman said:
Welcome to PF.

Can you show us your work so far? We are not allowed to help you until we see your initial work. Thanks.
Sure, If I do a dsolve in Maple on the IVP I get the following solution seen in the screenshot. But how should I got about finding the maximum size of v? If T(40)=37? Thats what I am unsure on.
1649266637895.png
 

1. What is an Initial Value Problem (IVP) coefficient?

An IVP coefficient is a numerical value that is multiplied by the highest derivative of a function in an initial value problem. It is used to find the maximum size of the solution to the problem.

2. Why is it important to find the maximum size of an IVP coefficient?

Finding the maximum size of an IVP coefficient is important because it helps determine the stability and accuracy of the solution to the problem. It also ensures that the solution does not exceed a certain threshold, which can lead to errors or instability in the system.

3. How is the maximum size of an IVP coefficient calculated?

The maximum size of an IVP coefficient is typically calculated using analytical or numerical methods. Analytical methods involve solving the problem algebraically, while numerical methods use computer algorithms to approximate the solution. Both methods involve finding the eigenvalues of the coefficient matrix.

4. What are some factors that can affect the maximum size of an IVP coefficient?

The maximum size of an IVP coefficient can be affected by the order of the derivative, the boundary conditions, the step size used in numerical methods, and the stability of the algorithm used to solve the problem.

5. How can the maximum size of an IVP coefficient be used in real-world applications?

The maximum size of an IVP coefficient is used in various fields of science and engineering, such as physics, chemistry, and electrical engineering. It is used to analyze the stability and accuracy of mathematical models and simulations, which can then be applied to real-world systems and processes.

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