Which of the following equation (s) is/are linear?

Options

A. `(dy)/(dx)+(y)/(x)=Inx`
B. `y((dy)/(dx))+4x=0`
C. `dx+dy=0`
D. `(d^(2)y)/(dx^(2))=cosx`

Answer

To determine which of the given differential equations are linear, we need to follow the definition of linearity in differential equations. A differential equation is considered linear if:

1. The exponent of the dependent variable (usually denoted as \(y\)) is 1.
2. The exponent of all derivatives (like \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\), etc.) is also 1.
3. There are no products of the dependent variable and its derivatives.

Now, let's analyze each option step by step.

### Step-by-Step Solution:

**Option 1:**
\[ \frac{dy}{dx} + \frac{y}{x} = \ln x \]
- The dependent variable is \(y\) with an exponent of 1.
- The derivative \(\frac{dy}{dx}\) has an exponent of 1.
- There is no product of \(y\) with its derivative.

**Conclusion:** This equation is linear.

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**Option 2:**
\[ \frac{dy}{dx} \cdot y + \frac{y}{x} = \ln x \]
- The dependent variable \(y\) has an exponent of 1.
- The derivative \(\frac{dy}{dx}\) has an exponent of 1.
- However, there is a product of the dependent variable \(y\) with its derivative \(\frac{dy}{dx}\).

**Conclusion:** This equation is non-linear.

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**Option 3:**
\[ dx + dy = 0 \]
- The dependent variable \(y\) has an exponent of 1.
- There are no derivatives of \(y\) present.

**Conclusion:** This equation is linear.

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**Option 4:**
\[ \frac{d^2y}{dx^2} + y = 0 \]
- The dependent variable \(y\) has an exponent of 1.
- The second derivative \(\frac{d^2y}{dx^2}\) has an exponent of 1.
- There is no product of \(y\) with its derivatives.

**Conclusion:** This equation is linear.

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### Final Summary:
The linear equations from the options are:
- Option 1
- Option 3
- Option 4

**Non-linear equation:**
- Option 2