# How do you solve this differential equation #dy/dx=(-x)/y# when #y=3# and #x=4# ?

# y^2 = 25 - x^2 #

We have:

Since this ODE can be separated, we can write:

Afterward, we are able to "separate the variables":

Afterward, we can easily integrate to obtain:

Thus, the Specific Resolution is:

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To solve the differential equation (\frac{dy}{dx} = -\frac{x}{y}) given that (y = 3) when (x = 4), we can use separation of variables.

Separating variables: [\frac{dy}{y} = -\frac{dx}{x}]

Integrating both sides: [\int \frac{1}{y} , dy = -\int \frac{1}{x} , dx]

[\ln|y| = -\ln|x| + C]

Where (C) is the constant of integration.

Rewriting in exponential form: [|y| = \frac{1}{|x|}e^C]

Given that (y = 3) when (x = 4), we can substitute these values into the equation and solve for (C): [3 = \frac{1}{4}e^C] [12 = e^C]

Taking the natural logarithm of both sides: [\ln(12) = C]

So the equation becomes: [|y| = \frac{1}{|x|} \cdot 12]

[y = \pm \frac{12}{x}]

Thus, the general solution to the differential equation is (y = \pm \frac{12}{x}).

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To solve the given differential equation ( \frac{dy}{dx} = -\frac{x}{y} ) when ( y = 3 ) and ( x = 4 ), we'll follow these steps:

- Substitute ( y = 3 ) and ( x = 4 ) into the equation to find the value of ( \frac{dy}{dx} ).
- Solve for ( \frac{dy}{dx} ) at ( y = 3 ) and ( x = 4 ).

( \frac{dy}{dx} = -\frac{x}{y} )

Substitute ( x = 4 ) and ( y = 3 ):

( \frac{dy}{dx} = -\frac{4}{3} )

Therefore, ( \frac{dy}{dx} = -\frac{4}{3} ) when ( y = 3 ) and ( x = 4 ).

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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.

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.

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.

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