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What is a solution to the differential equation #dy/dy=sqrt(1-y)#?

Answer 1

If this is written correctly, then you know that #(dy)/(dy) = 1#. That makes this really easy. But first, let's write this in the standard form:

#(dy)/(dy) - sqrt(1-y) = 0#

Now, let's use the fact that #(dy)/(dy) = 1# to get:

#1 - sqrt(1 - y) = 0#

#1 - (1 - y) = 0#

#=> color(blue)(y = 0)#

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Answer 2

Samantha Adkison

To solve the differential equation ( \frac{dy}{dx} = \sqrt{1-y} ), follow these steps:

Separate variables: Write the equation as ( \frac{1}{\sqrt{1-y}} , dy = dx ).
Integrate both sides: ( \int \frac{1}{\sqrt{1-y}} , dy = \int dx ).
Solve the integral: ( 2\sqrt{1-y} = x + C ), where ( C ) is the constant of integration.
Solve for ( y ): ( 1 - y = \left(\frac{x + C}{2}\right)^2 ).
Determine the constant: Use initial conditions if provided to find the specific value of ( C ).
Write down the solution: ( y = 1 - \left(\frac{x + C}{2}\right)^2 ).

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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.