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The order of the differential equation whose general solution is given by y = (c1 + c2) cos(x + c3) - c4ex + c5, where c1, c2, c3, c4 and c5 are arbitrary constants is
Correct answer is '3'. Can you explain this answer?
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The order of the differential equation whose general solution is give...
Explanation:
The given expression is the general solution of a differential equation. To determine the order of the differential equation, we need to find the highest order derivative present in the expression.
The given expression is: y = (c1 c2) cos(x c3) - c4ex c5
Let's break down the expression and identify the highest order derivative in each term:
Term 1: (c1 c2) cos(x c3)
- This term is a product of two constants c1 and c2, and the function cos(x c3).
- The function cos(x c3) has a maximum derivative of second order, as the second derivative of cos(x c3) is -cos(x c3).
- Therefore, the highest order derivative in this term is 2.
Term 2: - c4ex c5
- This term is the product of a constant c4, the function ex, and the constant c5.
- The function ex has a maximum derivative of first order, as the first derivative of ex is ex.
- Therefore, the highest order derivative in this term is 1.
Now, let's compare the highest order derivatives in each term:
- The highest order derivative in term 1 is 2.
- The highest order derivative in term 2 is 1.
Since the highest order derivative in term 1 is greater than the highest order derivative in term 2, the highest order derivative in the given expression is 2.
Therefore, the order of the differential equation whose general solution is given by y = (c1 c2) cos(x c3) - c4ex c5 is 2.
The given expression is the general solution of a differential equation. To determine the order of the differential equation, we need to find the highest order derivative present in the expression.
The given expression is: y = (c1 c2) cos(x c3) - c4ex c5
Let's break down the expression and identify the highest order derivative in each term:
Term 1: (c1 c2) cos(x c3)
- This term is a product of two constants c1 and c2, and the function cos(x c3).
- The function cos(x c3) has a maximum derivative of second order, as the second derivative of cos(x c3) is -cos(x c3).
- Therefore, the highest order derivative in this term is 2.
Term 2: - c4ex c5
- This term is the product of a constant c4, the function ex, and the constant c5.
- The function ex has a maximum derivative of first order, as the first derivative of ex is ex.
- Therefore, the highest order derivative in this term is 1.
Now, let's compare the highest order derivatives in each term:
- The highest order derivative in term 1 is 2.
- The highest order derivative in term 2 is 1.
Since the highest order derivative in term 1 is greater than the highest order derivative in term 2, the highest order derivative in the given expression is 2.
Therefore, the order of the differential equation whose general solution is given by y = (c1 c2) cos(x c3) - c4ex c5 is 2.
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Community Answer
The order of the differential equation whose general solution is give...
The given equation is
y = (c1 + c2) cos(x + c3) - c4ex + c5
Let y = A cos(x + B) - Cex; where A = c1 + c2, B = c3 and C = c4ec5
dy/dx = -A sin(x + B) - Cex
Differentiating again,
d2y/dx2 = -A cos(x + B) - Cex
Or d2y/dx2 + y= -2 Cex
Or d3y/dx3 + dy/dx = -2Cex = [d2y/dx2] + y
Or d3y/dx3 - d2y/dx2 + dy/dx - y = 0; which is a differential equation of order 3.
Alternate Solution:
y = (c1 + c2) cos(x + c3) + (-c4 ec5)ex
Put c1 + c2 = k1, c3 = k2, and -c4 ec5 = k3
Then, y = k1 cos (x + k2) + k3ex; where k1, k2 and k3 are constants.
Now, y contains 3 independent constants; hence, the order of the differential equation is 3.
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The order of the differential equation whose general solution is given by y = (c1 + c2) cos(x + c3) - c4ex + c5, where c1, c2, c3, c4 and c5 are arbitrary constants isCorrect answer is '3'. Can you explain this answer?
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