Find the differential equation by eliminating the constants a and b from the equation y = eˣ(a+bx²).

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Q. Find the differential equation by eliminating the constants a and b from the equation y = eˣ(a+bx²).

West Bengal WbchseWest Bengal HS (WBCHSE) Board 2025Subjective· 2mImportance★★★★★

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Differentiate twice to get two equations, then eliminate $a$ and $b$ between them.

Given $y=e^x(a+bx^2)$ , two constants means we need two derivatives to eliminate them.

First derivative:

$y' = e^x(a+bx^2) + e^x(2bx) = y + 2bx\,e^x \;\Rightarrow\; y'-y = 2bx\,e^x \quad(\ast)$

Second derivative:

$y'' = y' + 2b\,e^x + 2bx\,e^x = y' + 2b\,e^x + (y'-y)$

$\Rightarrow\; y''-2y'+y = 2b\,e^x \quad(\ast\ast)$

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Q.Find the differential equation by eliminating the constants a and b from the equation y = eˣ(a+bx²).