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The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is
(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)
(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x
(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2
(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6?
(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)
(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x
(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2
(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6?
Most Upvoted Answer
The solution of the second order differential equa- tion (d ^ 2 * y)/(...
Understanding the Differential Equation
The given second-order differential equation is:
(d^2 * y)/(d * x^2) - 2 * (dy/dx) - 3y = 6
This equation can be solved by finding the complementary solution (homogeneous part) and a particular solution (non-homogeneous part).
Step 1: Finding the Complementary Solution
1. Homogeneous Equation:
We first solve the homogeneous equation:
(d^2 * y)/(d * x^2) - 2 * (dy/dx) - 3y = 0
2. Characteristic Equation:
The characteristic equation is:
r^2 - 2r - 3 = 0
3. Roots Calculation:
Factoring gives:
(r - 3)(r + 1) = 0
Hence, r1 = 3 and r2 = -1.
4. Complementary Solution:
The complementary solution will be:
y_c = c1 * e^(-x) + c2 * e^(3x)
Step 2: Finding the Particular Solution
1. Particular Solution Guess:
For the constant term on the right (6), we can guess a particular solution:
y_p = A (where A is a constant).
2. Substituting into the Equation:
Substituting y_p into the original equation leads us to find A = -2.
3. Particular Solution:
Thus, y_p = -2.
Step 3: General Solution
Combining both solutions gives us the general solution:
y = y_c + y_p = c1 * e^(-x) + c2 * e^(3x) - 2.
Final Answer
Among the provided options, the correct answer is:
(c) c1 * e^(-x) + c2 * e^(3x) - 2.
The given second-order differential equation is:
(d^2 * y)/(d * x^2) - 2 * (dy/dx) - 3y = 6
This equation can be solved by finding the complementary solution (homogeneous part) and a particular solution (non-homogeneous part).
Step 1: Finding the Complementary Solution
1. Homogeneous Equation:
We first solve the homogeneous equation:
(d^2 * y)/(d * x^2) - 2 * (dy/dx) - 3y = 0
2. Characteristic Equation:
The characteristic equation is:
r^2 - 2r - 3 = 0
3. Roots Calculation:
Factoring gives:
(r - 3)(r + 1) = 0
Hence, r1 = 3 and r2 = -1.
4. Complementary Solution:
The complementary solution will be:
y_c = c1 * e^(-x) + c2 * e^(3x)
Step 2: Finding the Particular Solution
1. Particular Solution Guess:
For the constant term on the right (6), we can guess a particular solution:
y_p = A (where A is a constant).
2. Substituting into the Equation:
Substituting y_p into the original equation leads us to find A = -2.
3. Particular Solution:
Thus, y_p = -2.
Step 3: General Solution
Combining both solutions gives us the general solution:
y = y_c + y_p = c1 * e^(-x) + c2 * e^(3x) - 2.
Final Answer
Among the provided options, the correct answer is:
(c) c1 * e^(-x) + c2 * e^(3x) - 2.
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The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6?
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The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6? 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 second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6?.
The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6? 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 second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6? covers all topics & solutions for GATE Physics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6?.
Solutions for The solution of the second order differential equa- tion (d ^ 2 * y)/(d * x ^ 2) - 2 * dy/dx - 3y = 6 is(a) c_{1} * e ^ (- x) + c_{2} * e ^ (3x)(b) c_{1} * e ^ (-3x) + c_{2} * e ^ x(c) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) - 2(d) c_{1} * e ^ (- x) + c_{2} * e ^ (3x) + 6? in English & in Hindi are available as part of our courses for GATE Physics.
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