Question concerning constants of integration

This is because the expression (t^2 + 1) is always positive, so the argument of the logarithm will always be positive. Therefore, in summary, the expression y=\pm\sqrt{ln(t^{2}+1)+C} can be simplified to y=\pm\sqrt{ln[k(t^{2}+1)]}, where k is any positive real number.
  • #1
Duderonimous
63
1

Homework Statement



How can one simply let C=lnk? Thus changing

y=[itex]\pm[/itex][itex]\sqrt{ln(t^{2}+1)+C}[/itex]

to

y=[itex]\pm[/itex][itex]\sqrt{ln[k(t^{2}+1)]}[/itex]

Homework Equations



None

The Attempt at a Solution



I know they are both arbitrary constants, are there restrictions on the allowed values of the constants? Actually I checked the answer in the book and it said k is allowed to be any positive real number. I understand because it is under the radical. Insight into this would be helpful.
 
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  • #2
Duderonimous said:
I know they are both arbitrary constants, are there restrictions on the allowed values of the constants? Actually I checked the answer in the book and it said k is allowed to be any positive real number. I understand because it is under the radical. Insight into this would be helpful.

For every real [itex]C[/itex], there exists a unique [itex]k > 0[/itex] such that [itex]C = \ln k[/itex]. Thus one can always replace an arbitrary constant [itex]C[/itex] with [itex]\ln k[/itex] where [itex]k > 0[/itex] is arbitrary.
 

FAQ: Question concerning constants of integration

1. What are constants of integration?

Constants of integration are arbitrary numbers that are added to the solution of a differential equation to account for all possible solutions. They are typically represented by the letter "C" and are necessary because the derivative of a constant is always zero.

2. Why are constants of integration important?

Constants of integration are important because they allow for the inclusion of all possible solutions to a differential equation. Without them, the solution would be incomplete and would not accurately represent the problem at hand.

3. How do I determine the value of a constant of integration?

The value of a constant of integration cannot be determined solely from the differential equation itself. It must be determined by using additional information, such as initial conditions or boundary conditions, that are given in the problem.

4. Can there be more than one constant of integration in a solution?

Yes, it is possible for a solution to have more than one constant of integration. This is usually the case when a higher-order differential equation is being solved.

5. Are constants of integration always represented by the letter "C"?

No, constants of integration can be represented by any letter or symbol. The use of "C" is simply a convention that is commonly used in mathematics and science.

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