Differential Equations 1 Question 19
19.
Let $f: R \rightarrow R$ 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^{\prime}(x)=f(x) $
$\text { If } f(x) \neq 0 $
$\Rightarrow \frac{f^{\prime}(x)}{f(x)}=1 \Rightarrow \log f(x)=x+c $
$\Rightarrow f(x)=e^{c} e^{x} $
$\because f(0)=0 \Rightarrow e^{c}=0, \text { a contradiction } $
$\therefore 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 \quad \text { and } \quad f^{\prime}(x)=f(x) $
If $f(x) \neq 0$
$\Rightarrow \frac{f^{\prime}(x)}{f(x)}=1 \Rightarrow \ln f(x)=x+c $
$\Rightarrow f(x)=e^{c} \cdot e^{x} $
$\because f(0)=0$
$\Rightarrow e^{c}=0, a$ contradiction
$\therefore f(x)=0, \forall x \in R $
$\Rightarrow f(\ln 5)=0$
