SATHEE: Differential Equations 1 Question 19

Differential Equations 1 Question 19

19.

Let $f : R \to R$ be a continuous function, which satisfies $f (x) = \int_{0}^{x} f (t) d t$ . Then, the value of $f (\ln 5)$ is … .

(2009)

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Answer:

Correct Answer: 19. (0)

Solution:

From given integral equation, $f (0) = 0$ .

Also, differentiating the given integral equation w.r.t. $x$

$= f^{'} (x) = f (x)$

$If f (x) \neq 0$

$\Rightarrow \frac{f^{'} (x)}{f (x)} = 1 \Rightarrow \log f (x) = x + c$

$\Rightarrow f (x) = e^{c} e^{x}$

$∵ f (0) = 0 \Rightarrow e^{c} = 0, a contradiction$

$∴ f (x) = 0, \forall x \in R$

$\Rightarrow f (\ln 5) = 0$

Alternate Solution

Given,

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

$\Rightarrow$ $f (0) = 0 and f^{'} (x) = f (x)$

If $f (x) \neq 0$

$\Rightarrow \frac{f^{'} (x)}{f (x)} = 1 \Rightarrow \ln f (x) = x + c$

$\Rightarrow f (x) = e^{c} \cdot e^{x}$

$∵ f (0) = 0$

$\Rightarrow e^{c} = 0, a$ contradiction

$∴ f (x) = 0, \forall x \in R$

$\Rightarrow f (\ln 5) = 0$

Differential Equations 1 Question 19

19.

Answer:

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Differential Equations 1 Question 19

19.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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