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pav-90 [236]
3 years ago
15

A. Solve the differential equation

="y'=2x \sqrt{1-y^2} " alt="y'=2x \sqrt{1-y^2} " align="absmiddle" class="latex-formula">.
b. Explain why the initial value problem y'=2x \sqrt{1-y^2} with y(0) = 3 does not have a solution.
Mathematics
1 answer:
kirill [66]3 years ago
5 0
y' = \frac{dy}{dx}

seperable differential equations will have the form
\frac{dy}{dx} = F(x) G(y)

what you do from here is isolate all the y terms on one side and all the X terms on the other
\frac{dy}{G(y)} = F(x) dx
just divided G(y) to both sides and multiply dx to both sides

then integrate both sides
\int \frac{1}{G(y)} dy = \int F(x) dx



once you integrate, you will have a constant. use the initial value condition to solve for the constant, then try to isolate x or y if the question asks for it


In your problem,
G(y) = \sqrt{1-y^2}

F(x) = 2x

so all you need to integrate is
\int \frac{1}{\sqrt{1-y^2}} dy = \int 2x dx
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we suggest this system of equations:

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we solve this system of equation by substitution method.
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x=-20

y=x-3=-20-3=-23

Teh sum of the numerator an denominator=-20+(-23)=-43

Answer:    -43



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