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