Solving Diff Eqns: Renaming Constants & Reversing Signs

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In summary: This is similar to how you can combine like terms in algebra. However, when you negate a constant, you are changing its value and this may not always be valid. In this case, your book kept the negative sign because it is a more accurate representation of the original constant.
  • #1
Joseph1739
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Suppose I'm at this point in solving a differential equation and the initial condition is Q(0) = Q0
-ln|25-Q| + c1 = rt/100 + c2
Then if I combine c2-c1, I can rename it to c, we have:
-ln|25-Q| = rt/100 + c
Now if I multiply the equation by (-1), I get:
ln|25-Q| = -rt/100 - c
If I let -c = C:
ln|25-Q| -rt/100 +C

But if I rewrite -c = C, all the signs are reversed when I solve for Q. Also solving for the constant, my book kept the -c, and got c = Q0-25, but when I rewrite -c to C, I get C = 25-Q0.

So my question is, when can I rename constants? When I combined two constants it is okay to to rename the constant, but why is it incorrect when I negate a constant and rename it?
 
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  • #2
Joseph1739 said:
Suppose I'm at this point in solving a differential equation and the initial condition is Q(0) = Q0
-ln|25-Q| + c1 = rt/100 + c2
You can simplify things a bit by including the constant only on one side (the right side).

Joseph1739 said:
Then if I combine c2-c1, I can rename it to c, we have:
-ln|25-Q| = rt/100 + c
Now if I multiply the equation by (-1), I get:
ln|25-Q| = -rt/100 - c
If I let -c = C:
ln|25-Q| -rt/100 +C
You lost an = in the line above.
Joseph1739 said:
But if I rewrite -c = C, all the signs are reversed when I solve for Q. Also solving for the constant, my book kept the -c, and got c = Q0-25, but when I rewrite -c to C, I get C = 25-Q0.

So my question is, when can I rename constants?
Whenever you want to.
Joseph1739 said:
When I combined two constants it is okay to to rename the constant, but why is it incorrect when I negate a constant and rename it?
If -c = C, and the book shows c = ##Q_0 - 25##, then C = ##-(Q_0 - 25) = 25 - Q_0##.
 
  • #3
Joseph1739 said:
Suppose I'm at this point in solving a differential equation and the initial condition is Q(0) = Q0
-ln|25-Q| + c1 = rt/100 + c2
Then if I combine c2-c1, I can rename it to c, we have:
-ln|25-Q| = rt/100 + c
Now if I multiply the equation by (-1), I get:
ln|25-Q| = -rt/100 - c
If I let -c = C:
ln|25-Q| -rt/100 +C

But if I rewrite -c = C, all the signs are reversed when I solve for Q. Also solving for the constant, my book kept the -c, and got c = Q0-25, but when I rewrite -c to C, I get C = 25-Q0.

So my question is, when can I rename constants? When I combined two constants it is okay to to rename the constant, but why is it incorrect when I negate a constant and rename it?

Check your math. Your calculation of c and -C are wrong.

Looking at your last 2 equations, at [itex] t=0 [/itex] you either get [itex]ln\left|25-Q_0 \right| = -c [/itex] or [itex]ln\left|25-Q_0 \right| = C [/itex]. These two results agree with your definition [itex]c=-C [/itex].

You can always define a new constant as a combination of multiple constants.
 

1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time, and they are often used to model physical phenomena in science and engineering.

2. Why do constants need to be renamed and signs reversed when solving differential equations?

Constants need to be renamed and signs reversed in order to accurately solve the differential equation and find a specific solution. This step helps to eliminate any potential confusion and ensure that the final solution is correct.

3. Can differential equations be solved without renaming constants and reversing signs?

In some cases, differential equations can be solved without renaming constants and reversing signs. However, this approach may lead to incorrect solutions or make the problem more difficult to solve. It is generally recommended to follow the standard steps, including renaming constants and reversing signs, for more accurate and efficient solutions.

4. What are some common methods for solving differential equations?

Some common methods for solving differential equations include separation of variables, substitution, and using integrating factors. Different methods may be more suitable for solving certain types of differential equations, and it is important to use the appropriate method for each problem.

5. How are differential equations used in science?

Differential equations are used in science to model and understand various physical phenomena, such as motion, growth, and decay. They are also used to make predictions and analyze data in fields such as physics, chemistry, biology, and engineering.

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