Let f1, f2 and f3 be three curves satisfying the differential equation (1 – y2)dx = 2xy dy and pass through (0, 2). If f3 cuts the curves f1 and f2 at A and B respectively and one of the curve is passing through C(5,

Engineering

Mathematics

Variable Separable Differential Equation

Question

Let f₁, f₂ and f₃ be three curves satisfying the differential equation (1 – y²)dx = 2xy dy and pass through (0, 2). If f₃ cuts the curves f₁ and f₂ at A and B respectively and one of the curve is passing through C(5, –1), then find the area of ΔABC.

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$∵$ Given differential equation can be written as
$\int \frac{dx}{x} = \int \frac{2 y}{1 - y^{2}} dy = \int \frac{2 y}{y^{2} - 1} dy$
⇒ ln x = – ln (y2 – 1) + ln c
⇒ x (y2 – 1) = c
(0, 2) is on it ⇒ c = 0
$∴$ x (y² – 1) = 0 ⇒ x = 0, y = ±1
$Area of ΔABC = \frac{1}{2} \times 2 \times 5 = 5$