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39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is
(a) y^ prime prime - 4y + 4y = 0
(b) y deg - 5y + 6y = 0
(c) y - 4y = 0
(d) y^ prime prime - 3 * y' + 2y = 0?
(a) y^ prime prime - 4y + 4y = 0
(b) y deg - 5y + 6y = 0
(c) y - 4y = 0
(d) y^ prime prime - 3 * y' + 2y = 0?
Most Upvoted Answer
39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd orde...
Analysis of Given Particular Solutions
- We are given two particular solutions of a 2nd order homogeneous differential equation with constant coefficients: e^(2x) and x * e^(2x).
- Let's denote the unknown solution of the differential equation as y(x).
Deriving the Differential Equation
- The general form of a 2nd order homogeneous differential equation with constant coefficients is: y'' + ay' + by = 0.
Deriving the First Derivative
- The first derivative of e^(2x) is 2e^(2x) and the first derivative of x * e^(2x) is 2e^(2x) + 2xe^(2x).
Deriving the Second Derivative
- The second derivative of e^(2x) is 4e^(2x) and the second derivative of x * e^(2x) is 4e^(2x) + 4e^(2x) + 4xe^(2x).
Substituting into the General Form
- Substituting the first and second derivatives of y(x) into the general form of the differential equation, we get:
4e^(2x) + a(2e^(2x) + 2xe^(2x)) + b(e^(2x) + xe^(2x)) = 0.
Solving for a and b
- Simplifying the equation above, we get:
(4 + 2a + b)e^(2x) + (4a + b)e^(2x) = 0.
- Comparing the coefficients of e^(2x) and xe^(2x) to the particular solutions, we find that a = -3 and b = 2.
Final Differential Equation
- Therefore, the differential equation is: y'' - 3y' + 2y = 0.
Conclusion
- The correct answer is:
(d) y'' - 3y' + 2y = 0.
- We are given two particular solutions of a 2nd order homogeneous differential equation with constant coefficients: e^(2x) and x * e^(2x).
- Let's denote the unknown solution of the differential equation as y(x).
Deriving the Differential Equation
- The general form of a 2nd order homogeneous differential equation with constant coefficients is: y'' + ay' + by = 0.
Deriving the First Derivative
- The first derivative of e^(2x) is 2e^(2x) and the first derivative of x * e^(2x) is 2e^(2x) + 2xe^(2x).
Deriving the Second Derivative
- The second derivative of e^(2x) is 4e^(2x) and the second derivative of x * e^(2x) is 4e^(2x) + 4e^(2x) + 4xe^(2x).
Substituting into the General Form
- Substituting the first and second derivatives of y(x) into the general form of the differential equation, we get:
4e^(2x) + a(2e^(2x) + 2xe^(2x)) + b(e^(2x) + xe^(2x)) = 0.
Solving for a and b
- Simplifying the equation above, we get:
(4 + 2a + b)e^(2x) + (4a + b)e^(2x) = 0.
- Comparing the coefficients of e^(2x) and xe^(2x) to the particular solutions, we find that a = -3 and b = 2.
Final Differential Equation
- Therefore, the differential equation is: y'' - 3y' + 2y = 0.
Conclusion
- The correct answer is:
(d) y'' - 3y' + 2y = 0.
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39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0?
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39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0?.
39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about 39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0?.
Solutions for 39. If e ^ (2x) and x * e ^ (2x) are particular solution of a 2nd order differential equation (homogeneous) with constant co- efficients, then the equation is(a) y^ prime prime - 4y + 4y = 0(b) y deg - 5y + 6y = 0(c) y - 4y = 0(d) y^ prime prime - 3 * y' + 2y = 0? in English & in Hindi are available as part of our courses for UPSC.
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