GATE CS 2014 - Set 1 - 2 Marks Question
Duration: 2 min
This video lesson is available to enrolled students.
AI Summary
An AI-generated summary of this video lecture.
The video is a tutorial solving a calculus problem from the GATE CS 2014 exam. The problem asks to find the value of 't' given that the function f(x) = x * sin(x) satisfies the differential equation f''(x) + f(x) + t cos(x) = 0. The instructor systematically differentiates the function twice to find the second derivative, substitutes the expressions back into the original equation, simplifies the terms, and solves for the unknown constant 't'. The final answer is identified as -2. This is a classic example of verifying a function against a differential equation, a common topic in engineering mathematics.
Chapters
0:00 – 2:00 00:00-02:00
This window covers the core problem-solving process. It begins with the problem statement visible on screen: "Q. The function f(x) = x * sin(x) satisfies the following equation f''(x) + f(x) + t cosx = 0." The instructor writes the function f(x) = x sin x on the whiteboard. He then calculates the first derivative f'(x) = x cos x + sin x using the product rule. Next, he computes the second derivative f''(x) = -x sin x + cos x + cos x, which simplifies to 2 cos x - x sin x. He substitutes these values into the given equation: (2 cos x - x sin x) + (x sin x) + t cos x = 0. The x sin x terms cancel out, leaving 2 cos x + t cos x = 0. He factors this to (2 + t) cos x = 0, leading to the conclusion that t = -2. Finally, he circles option B on the multiple-choice list.
2:00 – 2:19 02:00-02:19
This short window shows the conclusion of the video. The instructor has finished solving the problem and is likely wrapping up the explanation. He points to the correct option B. -2 on the screen. The screen then transitions to a black background with the text "THANKS FOR WATCHING" in white, stylized font, signaling the end of the lecture segment.
The lecture demonstrates a standard application of differentiation rules, specifically the product rule, to solve a differential equation problem. By finding the second derivative of the given function and substituting it into the provided equation, the variable 't' is isolated and solved algebraically. This reinforces the connection between function properties and differential equations. The step-by-step derivation ensures clarity for students preparing for competitive exams like GATE.