The order and degree of the differential equation `((d^2y)/(dx^2))^(1/3)+(x+"dy"/"dx")^(1//2)` =0 are

Options

A. O=2 , D=2
B. O=2, D=`1/3`
C. `O=1/2`, D=2
D. O=2 , D=3

Answer

To determine the order and degree of the given differential equation

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}} + \left(x + \frac{dy}{dx}\right)^{\frac{1}{2}} = 0, \]

we will follow these steps:

### Step 1: Identify the highest derivative
The highest derivative present in the equation is \(\frac{d^2y}{dx^2}\), which is the second derivative of \(y\).

### Step 2: Determine the order of the differential equation
The order of a differential equation is defined as the highest order of derivative present in the equation. Since the highest derivative here is \(\frac{d^2y}{dx^2}\), the order of the differential equation is **2**.

### Step 3: Rewrite the equation in a standard form
To determine the degree, we need to express the equation in a polynomial form in terms of the derivatives. We can isolate the term with the second derivative:

\[ \left(\frac{d^2y}{dx^2}\right)^{\frac{1}{3}} = -\left(x + \frac{dy}{dx}\right)^{\frac{1}{2}}. \]

### Step 4: Eliminate the roots
To eliminate the fractional powers, we can cube both sides:

\[ \left(\frac{d^2y}{dx^2}\right) = -\left(x + \frac{dy}{dx}\right)^{\frac{3}{2}}. \]

### Step 5: Determine the degree of the differential equation
The degree of a differential equation is defined as the power of the highest derivative when the equation is expressed as a polynomial in derivatives. In this case, the highest derivative \(\frac{d^2y}{dx^2}\) appears to the power of 1 (since we have \(\left(\frac{d^2y}{dx^2}\right) = -\left(x + \frac{dy}{dx}\right)^{\frac{3}{2}}\)), which means the degree is **1**.

### Final Answer
Thus, the order of the differential equation is **2** and the degree is **1**.

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