Differential equation boundary conditions

In summary, differential equation boundary conditions are mathematical constraints applied at the boundaries of a domain to determine the behavior of the solution. They are important as they ensure a unique and accurate solution, and can be classified as initial conditions or boundary value conditions. These conditions can significantly affect the shape and behavior of the solution and can be adjusted during calculations, but with caution as it may require further calculations.
  • #1
kasse
384
1
[tex]
\frac{dN}{dt}=-k_sN^2
[/tex]

Attempt:

[tex]
\frac{1}{N^2}dN = -k_s dt
[/tex]

Integrate:

[tex]
-\frac{1}{N} + C = -k_s t
[/tex]

In the solution manual, C is written [tex]\frac{1}{N_0}[/tex]

Why?
 
Last edited:
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  • #2
If you integrate from 0 to t on the right, you have to integrate from N0 to Nt on the left, where N0 is the value of N at t = 0.
 
  • #3
Alternatively use the boundary conditions in this case [itex]N(0)=N_0[/itex] and then solve for C.
 

FAQ: Differential equation boundary conditions

What are differential equation boundary conditions?

Differential equation boundary conditions are mathematical conditions that are applied at the boundaries of a domain in which a differential equation is being solved. These conditions are used to specify the behavior of the solution at the boundaries and are essential for obtaining a unique solution to the differential equation.

Why are boundary conditions important in differential equations?

Boundary conditions are important in differential equations because they provide constraints that help determine the behavior of the solution at the boundaries of the domain. Without boundary conditions, the solution to a differential equation may not be unique and may not accurately represent the physical system being modeled.

What are the types of boundary conditions in differential equations?

The types of boundary conditions in differential equations can be classified as either initial conditions or boundary value conditions. Initial conditions specify the behavior of the solution at a single point within the domain, while boundary value conditions specify the behavior of the solution at the boundaries of the domain.

How do boundary conditions affect the solution to a differential equation?

Boundary conditions affect the solution to a differential equation by providing constraints on the behavior of the solution at the boundaries of the domain. These conditions can significantly impact the shape and behavior of the solution, and without them, the solution may not accurately reflect the physical system being modeled.

Can boundary conditions be changed or adjusted during a differential equation calculation?

Yes, boundary conditions can be changed or adjusted during a differential equation calculation. In some cases, changing the boundary conditions may be necessary to obtain a unique solution or to better represent the physical system being modeled. However, this should be done with caution, as it can significantly affect the overall solution and may require additional calculations.

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