Question 22 of 96intermediate🔧 ApplyShort Answer2 marks

In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is a solution of the corresponding differential equation: y=a2−x2x∈(−a,a)y = \sqrt{a^2 - x^2} x \in (-a, a) : x+ydydx=0(y≠0)x + y \frac{dy}{dx} = 0 (y \neq 0)

Correct Answer

Given function: y = a2−x2\sqrt{a² - x²}

Differentiating both sides with respect to x: dy/dx = (1/2)(a² - x²)^(-1/2) · (-2x) = -x/a2−x2\sqrt{a² - x²} = -x/y

Substituting y and dy/dx in the differential equation x + y(dy/dx) = 0:

L.H.S. = x + y · (-x/y) = x - x = 0 = R.H.S.

Since L.H.S. = R.H.S., the given function is a solution of the corresponding differential equation.

Exercise: EXERCISE 9.2 | Q: 10 | (Page 7)
For More Understanding

Explanation

This is a verification problem from Exercise 9.2. The student needs to differentiate the given function and substitute into the differential equation. The key step is recognizing that dy/dx = -x/y when y = a2−x2\sqrt{a² - x²}, which simplifies the substitution.

Solution Steps

  1. Step 1: Differentiate y = a2−x2\sqrt{a² - x²} to find dy/dx = -x/y

  2. Step 2: Substitute y and dy/dx into x + y(dy/dx) = 0

  3. Step 3: Verify L.H.S. = x + y(-x/y) = x - x = 0 = R.H.S.

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