dy/dx = x/y , y(0) = −3

-(x^2+9)^1/2

-3

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To solve this differential equation, we can use separation of variables.

First, let's rewrite the equation as:

y * dy = x * dx

Now we can integrate both sides of the equation with respect to their respective variables.

∫ y dy = ∫ x dx

Integrating the left side:

(1/2) * y^2 + C1 = (1/2) * x^2 + C2

Here, C1 and C2 are constants of integration.

Next, we'll solve for C1 using the given initial condition y(0) = -3.

(1/2) * (-3)^2 + C1 = (1/2) * 0^2 + C2

9/2 + C1 = C2

Now we can substitute C1 into our equation:

(1/2) * y^2 + (9/2 + C1) = (1/2) * x^2 + C2

(1/2) * y^2 + 9/2 = (1/2) * x^2 + 9/2 + C2

Simplifying the equation:

(1/2) * y^2 = (1/2) * x^2 + C2

Now, we can substitute the initial condition y(0) = -3 into our equation:

(1/2) * (-3)^2 = (1/2) * 0^2 + C2

(9/2) = (9/2) + C2

C2 = 0

Therefore, our equation becomes:

(1/2) * y^2 = (1/2) * x^2

Now we can solve for y:

y^2 = x^2

Taking the square root of both sides:

y = ±x

Now we can substitute the initial condition y(0) = -3 into our equation:

-3 = ±0

Since we can't have y = 0 and -3 at the same time, we can exclude the y = 0 solution.

Finally, the solution to the differential equation dy/dx = x/y with the initial condition y(0) = -3 is:

y = x