Trouble with needless of absolute value

[mech-eng]

Hi, while solving differential equations problems we have to sometimes use absolute value while
taking an integral of which 1 over a function. But in some problems I understand that we do not
have to use absolute value sign for example if the function related with atom number which can
never be a negative number we do not have to use absolute value. But in the picture I added here there is no sign about physical meaning of the problem so I think we should use absolute value and so there should have to be two different answers. The second picture of handwritting is
my other solution. The last one is a topic related with the problems. Have a nice day.

 

[HallsofIvy]

Equation 2.8 certainly should be $ln|y+ 1|= ln|x|+ cnst= ln|x|+ ln|a|= ln|ax|$

However, once we have gotten rid of the logarithms on both sides, we have y+ 1= ax, whether y, x, and a are positive or negative, and no longer need the absolute value signs.

 

[mech-eng]

Equation 2.8 certainly should be $ln|y+ 1|= ln|x|+ cnst= ln|x|+ ln|a|= ln|ax|$

However, once we have gotten rid of the logarithms on both sides, we have y+ 1= ax, whether y, x, and a are positive or negative, and no longer need the absolute value signs.

But in this situation y +1 might be positive whereas ax might negative so y + 1 might equal to -ax which leads to two different equalities, isn't it?

 

[HallsofIvy]

No, that can't happen. "a" is a constant determined by the initial value. If we are given an initial value $(x_0, y_0)$ such that $y_0+ 1$ and $x_0$ have the same sign, then a will be positive and y+ 1 and x will have the same sign for all x. If we are given an initial value $(x_0, y_0)$ such that $y_0+ 1$ and $x_0$ have opposite sign, then a is negative and y+1 and x will have opposite sign for all x. In either case, y+ 1 and ax will have the same sign

 

[blue_raver22]

Hello,

Thank you for sharing your thoughts on the use of absolute value in solving differential equations. I understand your concern about the potential for different answers when using or not using absolute value.

The use of absolute value in differential equations is often necessary when dealing with certain functions that can have negative values. However, as you pointed out, there are cases where the physical meaning of the problem dictates that absolute value is not needed.

In the case of the problem you shared, without any indication of the physical meaning, it is important to consider all possibilities and use absolute value to ensure the most accurate solution. However, if the problem was related to a physical system where negative values are not possible, then the use of absolute value may not be necessary.

It is important to carefully consider the physical context of a problem when deciding whether or not to use absolute value. In some cases, it may be helpful to consult with a colleague or mentor for their insights.

Overall, it is always important to double-check and make sure that the use of absolute value is appropriate for a given problem to avoid any potential errors. I hope this helps clarify the issue and I wish you all the best in your problem-solving endeavors. Have a great day.