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Q.

Let $f : R \to R$ be a function satisfying the differential equation $\frac{d y}{d x} + 2 x y = \sin^{2} x a n d f (0) = 0 .$ Then which of the following is correct ?

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a

$2 f (x) = e^{x}$ has at least one real solution.

b

$\lim_{x \to \infty} f (x) = \frac{1}{2}$

c

f is an even function.

d

$f (4) < \frac{1}{2}$

answer is D.

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Detailed Solution

Integrating factor $= e^{x^{2}}$

$∴ f (x) = e^{- x^{2}} \int_{0}^{x} e^{x^{2}} \sin^{2} x d x < e^{- x^{2}} \int_{0}^{x} e^{x^{2}} . x d x \forall x > 0$

$∴ f (x) < \frac{1}{2} \forall x > 0$

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