Question 5 of 96beginner💡 UnderstandVery Short1 mark

Determine order and degree (if defined) of differential equations given in Exercises 1 to 10.

d2ydx2=cos⁡3x+sin⁡3x\frac{d^2y}{dx^2} = \cos 3x + \sin 3x
Correct Answer

The highest order derivative present in the differential equation is d2ydx2\frac{d^2y}{dx^2}, so its order is two. It is a polynomial equation in d2ydx2\frac{d^2y}{dx^2} and the highest power raised to d2ydx2\frac{d^2y}{dx^2} is one, so its degree is one.

Exercise: EXERCISE 9.1 | Q: 5 | (Page 4)
For More Understanding

Explanation

Following the method in Example 1(i) of the provided context, we identify the highest derivative to determine the order. The degree is determined by checking if the equation is a polynomial in its derivatives and identifying the highest power of the highest order derivative. Since the equation is linear in the second derivative, the degree is 1.

Solution Steps

  1. Step 1: Identify the highest order derivative present in the equation d2ydx2=cos⁡3x+sin⁡3x\frac{d^2y}{dx^2} = \cos 3x + \sin 3x. The highest derivative is d2ydx2\frac{d^2y}{dx^2}, which is a second-order derivative. Therefore, the order is 2.

  2. Step 2: Determine the degree by checking if the equation is a polynomial in derivatives. The equation can be viewed as (d2ydx2)1−(cos⁡3x+sin⁡3x)=0\left(\frac{d^2y}{dx^2}\right)^1 - (\cos 3x + \sin 3x) = 0. Since it is a polynomial equation in d2ydx2\frac{d^2y}{dx^2} with the highest power being 1, the degree is 1.

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