Solution of the differential equation 3ydy/dx+2x=0 represents a family of
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... a family of ellipses centred at the origin.
The ellipses have eccentricity 1/√3 and a horizontal major axis.
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Step-by-step explanation:
Qualitatively:
dy/dx = -2x / 3y.
When x = 0, the curve is horizontal
When y = 0, the curve is vertical.
In the 1st and 3rd quadrants, the tangents have negative slope.
In the 2nd and 4th quadrants, the tangents have positive slope.
That matches an ellipse centred at the origin.
Precisely:
3ydy/dx + 2x = 0
=> 3y dy + 2x dx = 0
=> 3y² / 2 + x² = c²
=> y² / (c√(2/3))² + x² / c² = 1
This is an ellipse with minor semi-axis c√(2/3) and major semi-axis c.
The ratio between these is then √(2/3).
The eccentricity of the ellipse is
√( 1 - 2/3) = 1/√3.
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