What is the degree of the equation
`[(d^(2) y)/(dx^(2))] = [y + ((dy)/(dx))^(2) ]^(1/4)`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the degree of the given differential equation

\[ \frac{d^2 y}{dx^2} = \left( y + \left( \frac{dy}{dx} \right)^2 \right)^{\frac{1}{4}}, \]

we will follow these steps:

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

### Step 2: Rewrite the equation without fractional powers
To determine the degree, we need to eliminate any fractional powers. We can do this by raising both sides of the equation to the power of 4:

\[ \left( \frac{d^2 y}{dx^2} \right)^4 = y + \left( \frac{dy}{dx} \right)^2. \]

### Step 3: Identify the power of the highest order derivative
Now, we can see that the left-hand side of the equation has \(\left( \frac{d^2 y}{dx^2} \right)^4\). The power of the highest order derivative \(\frac{d^2 y}{dx^2}\) is 4.

### Step 4: Conclusion
Since the highest order derivative is raised to the power of 4, the degree of the given differential equation is 4.

Thus, the degree of the equation is:

\[ \text{Degree} = 4. \]

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