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The solution of the system of differential equations, [GATE 2001]
dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)
(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)
(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)
(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)
(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?
dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)
(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)
(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)
(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)
(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?
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The solution of the system of differential equations, [GATE 2001]dy/dx...
System of Differential Equations
To solve the system of differential equations given by:
1. dy/dx = y - z
2. dz/dx = -4y + z
we can approach it by finding the characteristic equation.
Step 1: Formulate the Matrix Representation
- Rewrite the system in matrix form:
d/dx [y, z] = [1, -1; -4, 1] [y; z]
Step 2: Calculate the Eigenvalues
- The characteristic polynomial is obtained from:
det(A - λI) = 0, where A = [1, -1; -4, 1]
- This leads to the characteristic equation:
λ^2 - 2λ + 3 = 0
- Solving this gives us eigenvalues λ = 3 and λ = -1.
Step 3: Find the Eigenvectors
- For λ = 3:
Substitute λ into (A - λI) to find the corresponding eigenvector.
- For λ = -1:
Similarly, substitute to find the eigenvector.
Step 4: General Solution
- The general solution is of the form:
y(x) = A * e^(3x) + B * e^(-x)
- Using the relationships derived from the original equations, we find:
z(x) = -2A * e^(3x) + 2B * e^(-x)
Final Verification
- Substitute the solutions back into the original equations to verify correctness.
Correct Answer
- The correct solution from the options provided is:
(a) y(x) = A * e^(3x) + B * e^(-x); z(x) = -2A * e^(3x) + 2B * e^(-x)
This solution satisfies both differential equations.
To solve the system of differential equations given by:
1. dy/dx = y - z
2. dz/dx = -4y + z
we can approach it by finding the characteristic equation.
Step 1: Formulate the Matrix Representation
- Rewrite the system in matrix form:
d/dx [y, z] = [1, -1; -4, 1] [y; z]
Step 2: Calculate the Eigenvalues
- The characteristic polynomial is obtained from:
det(A - λI) = 0, where A = [1, -1; -4, 1]
- This leads to the characteristic equation:
λ^2 - 2λ + 3 = 0
- Solving this gives us eigenvalues λ = 3 and λ = -1.
Step 3: Find the Eigenvectors
- For λ = 3:
Substitute λ into (A - λI) to find the corresponding eigenvector.
- For λ = -1:
Similarly, substitute to find the eigenvector.
Step 4: General Solution
- The general solution is of the form:
y(x) = A * e^(3x) + B * e^(-x)
- Using the relationships derived from the original equations, we find:
z(x) = -2A * e^(3x) + 2B * e^(-x)
Final Verification
- Substitute the solutions back into the original equations to verify correctness.
Correct Answer
- The correct solution from the options provided is:
(a) y(x) = A * e^(3x) + B * e^(-x); z(x) = -2A * e^(3x) + 2B * e^(-x)
This solution satisfies both differential equations.
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The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?
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The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? 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 The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?.
The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? 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 The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?.
Solutions for The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? in English & in Hindi are available as part of our courses for GATE Physics.
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Here you can find the meaning of The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? defined & explained in the simplest way possible. Besides giving the explanation of
The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)?, a detailed solution for The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? has been provided alongside types of The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? theory, EduRev gives you an
ample number of questions to practice The solution of the system of differential equations, [GATE 2001]dy/dx = y - z and dz/dx = - 4y + z is given by (for A and B are arbitrary constants)(a) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) + 2B * e ^ (- x)(b) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B ^ (- x)(c) y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = 2A * e ^ (3x) - 2B * e ^ (- x)(d y(x) = A * e ^ (3x) + B * e ^ (- x); z(x) = - 2A * e ^ (3x) - 2B * e ^ (- x)? tests, examples and also practice GATE Physics tests.
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