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Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA?
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Let f : R → R be a differentiable function with f 0 (x) = f(x) for all...
Given:
- f : R → R is a differentiable function with f 0 (x) = f(x) for all x.
- f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0.
- f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x).
To Find:
The value of p.
Solution:
Step 1: Find the second derivative of f(αx) and f(βx).
Differentiating f(αx) twice with respect to x, we get:
f''(αx) = (d/dx)(f'(αx))
= (d/d(αx))(f'(αx)) * (d(αx)/dx)
= α * f''(αx)
Similarly, differentiating f(βx) twice with respect to x, we get:
f''(βx) = β * f''(βx)
Step 2: Substitute the values of f(αx), f(βx), f''(αx), and f''(βx) into the differential equation.
Substituting f(αx) and f''(αx) into the differential equation, we get:
4 * α^2 * f''(αx) - p * α * f'(αx) + 3 * f(αx) = 0 ...(1)
Substituting f(βx) and f''(βx) into the differential equation, we get:
4 * β^2 * f''(βx) - p * β * f'(βx) + 3 * f(βx) = 0 ...(2)
Step 3: Simplify the given equations.
From equation (1), we can divide all terms by α^2:
4 * f''(αx) - (p * α)/α^2 * f'(αx) + (3 * f(αx))/α^2 = 0
4 * f''(αx) - (p/α) * f'(αx) + (3/α^2) * f(αx) = 0 ...(3)
From equation (2), we can divide all terms by β^2:
4 * f''(βx) - (p * β)/β^2 * f'(βx) + (3 * f(βx))/β^2 = 0
4 * f''(βx) - (p/β) * f'(βx) + (3/β^2) * f(βx) = 0 ...(4)
Step 4: Substitute x = 2x into equation (3) and x = x into equation (4).
Substituting x = 2x into equation (3), we get:
4 * f''(2x) - (p/2) * f'(2x) + (3/4) * f(2x) = 0 ...(5)
Substituting x = x into
- f : R → R is a differentiable function with f 0 (x) = f(x) for all x.
- f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0.
- f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x).
To Find:
The value of p.
Solution:
Step 1: Find the second derivative of f(αx) and f(βx).
Differentiating f(αx) twice with respect to x, we get:
f''(αx) = (d/dx)(f'(αx))
= (d/d(αx))(f'(αx)) * (d(αx)/dx)
= α * f''(αx)
Similarly, differentiating f(βx) twice with respect to x, we get:
f''(βx) = β * f''(βx)
Step 2: Substitute the values of f(αx), f(βx), f''(αx), and f''(βx) into the differential equation.
Substituting f(αx) and f''(αx) into the differential equation, we get:
4 * α^2 * f''(αx) - p * α * f'(αx) + 3 * f(αx) = 0 ...(1)
Substituting f(βx) and f''(βx) into the differential equation, we get:
4 * β^2 * f''(βx) - p * β * f'(βx) + 3 * f(βx) = 0 ...(2)
Step 3: Simplify the given equations.
From equation (1), we can divide all terms by α^2:
4 * f''(αx) - (p * α)/α^2 * f'(αx) + (3 * f(αx))/α^2 = 0
4 * f''(αx) - (p/α) * f'(αx) + (3/α^2) * f(αx) = 0 ...(3)
From equation (2), we can divide all terms by β^2:
4 * f''(βx) - (p * β)/β^2 * f'(βx) + (3 * f(βx))/β^2 = 0
4 * f''(βx) - (p/β) * f'(βx) + (3/β^2) * f(βx) = 0 ...(4)
Step 4: Substitute x = 2x into equation (3) and x = x into equation (4).
Substituting x = 2x into equation (3), we get:
4 * f''(2x) - (p/2) * f'(2x) + (3/4) * f(2x) = 0 ...(5)
Substituting x = x into
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Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA?
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Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA?.
Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA?.
Solutions for Let f : R → R be a differentiable function with f 0 (x) = f(x) for all x. Suppose that f(αx) and f(βx) are two non-zero solutions of the differential equation 4 d 2 y dx2 − p dy dx 3y = 0 satisfying f(αx)f(βx) = f(2x) and f(αx)f(−βx) = f(x). Then, the value of p is . MA? in English & in Hindi are available as part of our courses for Mathematics.
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