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The differential equation corresponding to the family of curves,
y = A.sin x + B.cos (x + C), where A, B, C are arbitrary constants, has the order ______
y = A.sin x + B.cos (x + C), where A, B, C are arbitrary constants, has the order ______
- a)2
- b)1
- c)0
- d)3
Correct answer is option 'A'. Can you explain this answer?
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The differential equation corresponding to the family of curves, y = A...
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The differential equation corresponding to the family of curves, y = A...
The given family of curves is represented by the equation y = A.sin(x + B.cos(C)), where A, B, and C are arbitrary constants.
To find the corresponding differential equation, we need to differentiate the given equation with respect to x.
Differentiating y = A.sin(x + B.cos(C)) with respect to x, we get:
dy/dx = A.cos(x + B.cos(C)) * (1 - B.sin(C))
The above expression represents the derivative of y with respect to x. To find the differential equation, we need to eliminate the constants A, B, and C from the expression.
Now, let's differentiate dy/dx = A.cos(x + B.cos(C)) * (1 - B.sin(C)) with respect to x:
d^2y/dx^2 = -A.sin(x + B.cos(C)) * (1 - B.sin(C)) + A.cos(x + B.cos(C)) * (-B.cos(C))
Simplifying the above expression, we get:
d^2y/dx^2 = A[cos(x + B.cos(C)) * (-B.cos(C)) - sin(x + B.cos(C)) * (1 - B.sin(C))]
Further simplifying, we get:
d^2y/dx^2 = -A[B.cos(x + B.cos(C)) * cos(C) + sin(x + B.cos(C)) * (B.sin(C) - 1)]
The expression obtained above represents the second derivative of y with respect to x. To determine the order of the differential equation, we count the highest order derivative. In this case, the highest order derivative is d^2y/dx^2.
Therefore, the order of the differential equation is 2, which corresponds to option 'A'.
To find the corresponding differential equation, we need to differentiate the given equation with respect to x.
Differentiating y = A.sin(x + B.cos(C)) with respect to x, we get:
dy/dx = A.cos(x + B.cos(C)) * (1 - B.sin(C))
The above expression represents the derivative of y with respect to x. To find the differential equation, we need to eliminate the constants A, B, and C from the expression.
Now, let's differentiate dy/dx = A.cos(x + B.cos(C)) * (1 - B.sin(C)) with respect to x:
d^2y/dx^2 = -A.sin(x + B.cos(C)) * (1 - B.sin(C)) + A.cos(x + B.cos(C)) * (-B.cos(C))
Simplifying the above expression, we get:
d^2y/dx^2 = A[cos(x + B.cos(C)) * (-B.cos(C)) - sin(x + B.cos(C)) * (1 - B.sin(C))]
Further simplifying, we get:
d^2y/dx^2 = -A[B.cos(x + B.cos(C)) * cos(C) + sin(x + B.cos(C)) * (B.sin(C) - 1)]
The expression obtained above represents the second derivative of y with respect to x. To determine the order of the differential equation, we count the highest order derivative. In this case, the highest order derivative is d^2y/dx^2.
Therefore, the order of the differential equation is 2, which corresponds to option 'A'.
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