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Explain why both y = x-3 and y = x+2 can both be considered solutions to the differential equation dy/dx - 1 = 0?

Answer 1

Samantha Adkison

See below.

A given function #g(x)# is a solution for the differential equation

#y'-1=0# if #g'-1=0# Now substituting #g(x) = x+C# into the differential equation we have

#d/(dx)(x+C)-1 = 1-1=0# so the solution

#y = x + C# includes #x-3# and #x+2# because #C# represents any constan in #RR#.

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Answer 2

Christopher Agresta

An alternative approach is to work backwards from the Differential Equation and provide the General Solution.

We have:

# dy/dx - 1 = 0 #

This is a First Order separable DE, and we can write in the form:

# dy/dx = 1 #

And "separating the variables" gives us:

# int y \ dx = int 1 \ dx #

Integrating we get the General Solution :

# y = x + C#

where #C# is an Arbitrary constant. Thus

# y=x-3 => C=-3 # # y=x+2 => C = -2 #

Are both solutions, as is for example:

# y=x+3 \ \ => C=3 # # y=x-90 => C=-90 # etc

Given an initial condition we can provide a value for the Arbitrary constant and supply a unique solution or a Particular Solution .

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.