Order of the differential equation `(d^(2)y)/(dx^(2))+5(dy)/(dx)+intydx=x^(3)` is

Options

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

Answer

To determine the order of the given differential equation

\[ \frac{d^2y}{dx^2} + 5\frac{dy}{dx} + \int y \, dx = x^3, \]

we will analyze the components of the equation step by step.

### Step 1: Identify the highest derivative present
The equation contains the following derivatives:
- \(\frac{d^2y}{dx^2}\) (second derivative)
- \(\frac{dy}{dx}\) (first derivative)
- The integral \(\int y \, dx\) is not a derivative but an integral of \(y\).

### Step 2: Understand the order of the differential equation
The order of a differential equation is defined as the highest order of derivative present in the equation. In our case, the highest derivative is \(\frac{d^2y}{dx^2}\), which is a second derivative.

### Step 3: Consider the integral term
The integral term \(\int y \, dx\) can be thought of as introducing an arbitrary constant when differentiated. When we differentiate the entire equation with respect to \(x\), we will introduce a new term related to the first derivative of \(y\).

### Step 4: Differentiate the equation
To find the order, we will differentiate the entire equation once:

\[ \frac{d}{dx}\left(\frac{d^2y}{dx^2} + 5\frac{dy}{dx} + \int y \, dx\right) = \frac{d}{dx}(x^3). \]

This gives us:

\[ \frac{d^3y}{dx^3} + 5\frac{d^2y}{dx^2} + y = 3x^2. \]

### Step 5: Identify the highest derivative after differentiation
Now, the highest derivative present in this new equation is \(\frac{d^3y}{dx^3}\), which is a third derivative.

### Conclusion
Thus, the order of the original differential equation is **3**.

### Final Answer
The order of the differential equation is **3**.

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