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)

Passage Based Problems

Passage

Let $f : [0, 1] \to R$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f (0) = f (1) = 0$ and satisfies

$f^{''} (x) - 2 f^{'} (x) + f (x) \geq e^{x}, x \in [0, 1]$

(2013 Adv.)

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Answer:

Correct Answer: 19. (c)

Solution:

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

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

$\begin{array}{lc} f^{'} (x) = f (x) \\ If & f (x) \neq 0 \\ \Rightarrow & \frac{f^{'} (x)}{f (x)} = 1 \Rightarrow \log f (x) = x + c \\ \Rightarrow & f (x) = e^{c} e^{x} \\ ∵ & f (0) = 0 \Rightarrow e^{c} = 0, a contradiction \\ ∴ & f (x) = 0, \forall x \in R \\ \Rightarrow & f (\ln 5) = 0 \end{array}$

Alternate Solution

Given,

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

$\Rightarrow$

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

If $f (x) \neq 0$

$\begin{array}{ll} \Rightarrow & \frac{f^{'} (x)}{f (x)} = 1 \Rightarrow \ln f (x) = x + c \\ \Rightarrow & f (x) = e^{c} \cdot e^{x} \\ ∵ & f (0) = 0 \end{array}$

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

$\begin{array}{lc} ∴ & f (x) = 0, \forall x \in R \\ \Rightarrow & f (\ln 5) = 0 \end{array}$

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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 … .

Passage Based Problems

Passage

Answer:

Alternate Solution

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Differential Equations 1 Question 19

19. Let f:R→R be a continuous function, which satisfies f(x)=∫0xf(t)dt. Then, the value of f(ln⁡5) is … .

Passage Based Problems

Passage

Answer:

Alternate Solution

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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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 … .