Differential equuation with boundary conditions

In summary: Homework Statement In summary, the problem involves finding the general solution to a differential equation with two boundary conditions. Using basic integration techniques, the equation is rearranged and solved for T. The resulting general solution is then used to find the constants C1 and C2 by plugging in the given boundary conditions.
  • #1
ookt2c
16
0

Homework Statement


d^2T/dx^2+S/K=0 Boundary Conditions T=Tsub1 @ x=0
and T=Tsub2 @ x=L


Homework Equations





The Attempt at a Solution



d^2T/dx^2 = -(S/K) <--- intergrate to get
dT=-(S/K)dx+ C1 <--- intergrate to get
T=(-S/K)x+c1+c2
apply both boundary conditions to get
Tsub1=c1+c2

Not sure if i doing it right and if i am i don't know how to get c1 and c2
Thanks for your help
Tsub2=-(S/K)*L +c1+c2
 
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  • #2
ookt2c said:
d^2T/dx^2 = -(S/K) <--- intergrate to get

You mean "integrate", right :wink: And you aren't integrating to get to here, you are simply rearranging your DE algebraically.

dT=-(S/K)dx+ C1 <--- intergrate to get

No,

[tex]\frac{d^2T}{dx^2}=-\frac{S}{K} \implies \frac{dT}{dx}=\int -\frac{S}{K}dx= -\frac{S}{K}x+C_1[/tex]

You will need to integrate once more to get [itex]T[/itex]
 
  • #3
I meant integrate that equation to get to the next line.

And after I integrate I get: T=-(S/K)* (x^2)/2+C1+C2

Now I just plug in the boudary conditions and solve the system of equations for c1 and c2 correct?
 
  • #4
ookt2c said:
I meant integrate that equation to get to the next line.

And after I integrate I get: T=-(S/K)* (x^2)/2+C1+C2

Now I just plug in the boudary conditions and solve the system of equations for c1 and c2 correct?

[tex]\int(-\frac{S}{K}x+C_1)dx\neq -\frac{S}{K}x^2+C_1 +C_2[/tex]

You're missing something.
 
  • #5
=-S/K* x^2/2+C1x+C2

Also how do I make my equations appear like yours?
 
  • #6
ookt2c said:
=-S/K* x^2/2+C1x+C2

Good, now use your boundary conditions to find C1 and C2

Also how do I make my equations appear like yours?

Click on the link in my sig :wink:
 

1. What is a differential equation with boundary conditions?

A differential equation with boundary conditions is a mathematical equation that involves an unknown function and its derivatives. The boundary conditions specify the values of the function at certain points or under certain conditions. These conditions are used to solve the equation and find the function that satisfies both the equation and the given boundary values.

2. What are the types of boundary conditions in differential equations?

The types of boundary conditions in differential equations are:

  1. Dirichlet boundary conditions: specify the values of the function at certain points in the domain.
  2. Neumann boundary conditions: specify the values of the derivative of the function at certain points in the domain.
  3. Robin boundary conditions: specify a linear combination of the function and its derivative at certain points in the domain.

3. How are boundary conditions used in solving differential equations?

Boundary conditions are used to find the particular solution of a differential equation. They are applied to the general solution, which is the solution without any specific values for the constants, to obtain the particular solution that satisfies both the equation and the given boundary values.

4. What is the significance of boundary conditions in real-world applications?

Boundary conditions are essential in real-world applications as they help in modeling and simulating physical phenomena. For example, in heat transfer problems, the temperature at the boundaries of a system is known, and this information is used as boundary conditions to determine the temperature distribution within the system.

5. Can boundary conditions be changed or modified?

Yes, boundary conditions can be changed or modified depending on the problem or situation. For instance, in a heat transfer problem, the boundary conditions can be changed if the material or environment of the system is altered. This would result in a different solution to the differential equation with the new boundary conditions.

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