Why can we combine constants of integration into C?
When we solve an integral or a differential equation, a line containing several constants often becomes a line with only C. We’re giving a new name to a quantity that is still constant. Let’s look at when that renaming keeps the same family of answers and how any restrictions carry through.
Why two integration constants can become one
Integrating both sides of a separable equation can give us a constant on each side. Suppose the integrated equation has the form
Subtract C₁ from both sides.
The difference C₂ − C₁ does not depend on x or y. Because the two original constants are arbitrary real numbers, their difference can be any real number too. We can name that difference C:
We have described the same family using one constant instead of two.
Example: Rename a negative constant
When we integrate on a branch where y is nonzero, we get
Multiply the entire equation by −1.
If C₁ can be any real number, then −C₁ can also be any real number. Define a new constant C₂ = −C₁ and write
This expression is defined on intervals where the denominator is nonzero. Renaming the constant did not change that condition or the nonzero-y branch we were working with.
What happens to e raised to a constant?
An expression can be constant without being allowed to take every real value. For example, eᴷ is positive for every real K. If we give eᴷ the name A, then A must remain positive.
If the algebra gives a choice of positive or negative sign, then we can write
That A can be any nonzero real number. Zero would need its own justification in the original equation; it is not produced by either sign of eᴷ.
Which terms can be absorbed into a constant?
A quantity such as can be renamed C when C₁ is an arbitrary real constant, because adding 4 still lets the result range over every real number. The same reasoning works for a nonzero constant multiple, such as .
A variable-dependent term stays variable-dependent. For example, in , we can combine C₁ + 4 into C and write , while the x² term stays in the expression.
So each time we rename a constant, we ask two things: does this quantity depend on the variable, and what values can it take? Keeping those answers intact is what makes the shorter notation represent the same result.