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Vector Algebra
Question
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Form the differential equation of the family of curves by eliminating arbitrary constants a and b.
y = e2x (a + b x)

Answer

The equation of family of curves is
y = e2x (a + b x)    ...(1)
Differentiating both sides w.r.t. x, we get,
y1 = e2x  b + (a + b x). 2 e2x
or    y1 = b e2x + 2 y    [∵ of(1)]
∴ y1 – 2 y = be2x        ...(2)
Again differentiating w.r.t. x, we get,
y2 – 2 y1 = 2 b e2x
or    y2 – 2 y1 = 2 (y1 – 2 y)    [∵ of (2)]
or    y2 – 2 y1 = 2 y1 – 4 y
or    y2 – 4 y1 + 4 y = 0, which is required differential equation.

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