Separable Equations

Question

Find an equation of the curve that passes through the point (0, 2) and whose slope at (x, y) is x/y.

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Answer 1

dy/dx = x/y

ydy = xdx

∫ ydy = ∫ xdx

(1/2)y² = (1/2)x² + C

Since (0,2) is on the curve, (1/2)(2²) = (1/2)(0)² + C. So, C = 2.

(1/2)y² = (1/2)x² + 2

y² = x² + 4

y = √ (x² + 4)