Let $f$ be a differentiable function satisfying $f(x) = \frac{2}{\sqrt{3}} \int_0^{\sqrt{3}} f\left(\frac{\lambda^2 x}{3}\right) d\lambda$, $x > 0$ and $f(1) = \sqrt{3}$. If $y = f(x)$ passes through the point $(\alpha, 6)$, then $\alpha$ is equal to ____.

JEE Main · 2022 · 27 Jul 2022 · Shift 2 · Differential Equations

Let

f

be a differentiable function satisfying

f (x) = \frac{2}{3} \int_{0}^{3} f (\frac{λ ^{2} x}{3}) d λ

x > 0

and

f (1) = 3

. If

y = f (x)

passes through the point

(α, 6)

, then

α

is equal to ____.

Answer: 12

Substituting

t = \frac{λ ^{2} x}{3}

and transforming the integral:

f (x) = \frac{1}{x} \int_{0}^{x} \frac{f ( t )}{t} d t

.

Differentiating both sides using Leibniz rule:

x f^{'} (x) + \frac{f ( x )}{2 x} = \frac{f ( x )}{x}

.

Simplifying:

x f^{'} (x) = \frac{f ( x )}{2 x}

, which gives

\frac{d y}{y} = \frac{d x}{2 x}

.

Integrating:

ln y = \frac{1}{2} ln x + c

, so

f (x) = C x

.

Using

f (1) = 3

C = 3

, thus

f (x) = 3 x

.

For

f (α) = 6

3 α = 6 \Rightarrow α = 12

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