GATE Physics Exam > GATE Physics Questions > Consider the differential equationdG(x)/dx+kG...
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Consider the differential equation
dG(x)/dx+kG(x) = delta (x)
where k is a constant. Which of the following state- ment is true?
(a) Both G(x) and G'(x) are continuous at x = 0
(b) G(x) is continuous at x = 0 but G'(x) is not.
(c) G(x) is discontinuous at x = 0
(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.?
dG(x)/dx+kG(x) = delta (x)
where k is a constant. Which of the following state- ment is true?
(a) Both G(x) and G'(x) are continuous at x = 0
(b) G(x) is continuous at x = 0 but G'(x) is not.
(c) G(x) is discontinuous at x = 0
(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.?
Most Upvoted Answer
Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is...
Understanding the Differential Equation
The given differential equation is:
dG(x)/dx + kG(x) = delta(x)
where delta(x) is the Dirac delta function, representing a point source at x = 0.
Analysis of G(x)
- The solution G(x) must account for the discontinuity introduced by the delta function at x = 0.
- For x ≠ 0, G(x) satisfies the homogeneous equation: dG(x)/dx + kG(x) = 0, leading to the general solution of the form G(x) = Ae^(-kx).
Behavior at x = 0
- The delta function implies that there is an impulse at x = 0, causing G(x) to have a jump discontinuity.
- Specifically, integrating the differential equation across an interval that includes x = 0 reveals that G(x) has a discontinuity in its derivative G'(x).
Continuity Properties
- G(x) is continuous at x = 0; however, G'(x) experiences a discontinuity due to the impulse effect of delta(x).
- Therefore, the following statements are evaluated:
- (a) Both G(x) and G'(x) are continuous at x = 0: False
- (b) G(x) is continuous at x = 0 but G'(x) is not: True
- (c) G(x) is discontinuous at x = 0: False
- (d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k: False
Conclusion
The correct answer is (b): G(x) is continuous at x = 0 but G'(x) is not. The discontinuity in the derivative arises from the nature of the delta function, affecting the behavior of the solution at that specific point.
The given differential equation is:
dG(x)/dx + kG(x) = delta(x)
where delta(x) is the Dirac delta function, representing a point source at x = 0.
Analysis of G(x)
- The solution G(x) must account for the discontinuity introduced by the delta function at x = 0.
- For x ≠ 0, G(x) satisfies the homogeneous equation: dG(x)/dx + kG(x) = 0, leading to the general solution of the form G(x) = Ae^(-kx).
Behavior at x = 0
- The delta function implies that there is an impulse at x = 0, causing G(x) to have a jump discontinuity.
- Specifically, integrating the differential equation across an interval that includes x = 0 reveals that G(x) has a discontinuity in its derivative G'(x).
Continuity Properties
- G(x) is continuous at x = 0; however, G'(x) experiences a discontinuity due to the impulse effect of delta(x).
- Therefore, the following statements are evaluated:
- (a) Both G(x) and G'(x) are continuous at x = 0: False
- (b) G(x) is continuous at x = 0 but G'(x) is not: True
- (c) G(x) is discontinuous at x = 0: False
- (d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k: False
Conclusion
The correct answer is (b): G(x) is continuous at x = 0 but G'(x) is not. The discontinuity in the derivative arises from the nature of the delta function, affecting the behavior of the solution at that specific point.
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Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.?
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Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.?.
Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.? for GATE Physics 2025 is part of GATE Physics preparation. The Question and answers have been prepared according to the GATE Physics exam syllabus. Information about Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.?.
Solutions for Consider the differential equationdG(x)/dx+kG(x) = delta (x)where k is a constant. Which of the following state- ment is true?(a) Both G(x) and G'(x) are continuous at x = 0(b) G(x) is continuous at x = 0 but G'(x) is not.(c) G(x) is discontinuous at x = 0(d) The continuity properties of G(x) and G'(x) at x = 0 depend on the value of k.? in English & in Hindi are available as part of our courses for GATE Physics.
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