Which one of the differential equation is not linear

Options

A. `(d^(2) y)/(dx^(2)) + 4y = 0`
B. `x"" (dy)/(dx) + y = x^(3)`
C. `(x- y)^(2) (dy)/(dx) = 9`
D. `cos^(2) x (dy)/(dx) + y = tan x `

Answer

To determine which differential equation is not linear, we need to analyze the given equations based on the criteria for linearity. A differential equation is considered linear if:

1. Every dependent variable and its derivatives occur only to the first degree.
2. There are no products of dependent variables or their derivatives.

Let's analyze the equations step by step.

### Step 1: Analyze the first differential equation
**Equation:** \(\frac{d^2y}{dx^2} + 4y = 0\)

- The highest derivative is \(\frac{d^2y}{dx^2}\), which is to the first degree (power of 1).
- The dependent variable \(y\) is also to the first degree.
- There are no products of \(y\) and its derivatives.

**Conclusion:** This equation is linear.

### Step 2: Analyze the second differential equation
**Equation:** \(x \cdot \frac{dy}{dx} + y = x\)

- The highest derivative is \(\frac{dy}{dx}\), which is to the first degree.
- The dependent variable \(y\) is also to the first degree.
- However, there is a product \(x \cdot \frac{dy}{dx}\), which involves the independent variable \(x\) and the derivative \(\frac{dy}{dx}\).

**Conclusion:** This equation is still considered linear because the product does not involve the dependent variable \(y\) itself.

### Step 3: Analyze the third differential equation
**Equation:** \((x - y)^2 \cdot \frac{dy}{dx} = 9\)

- Expanding \((x - y)^2\) gives us \(x^2 - 2xy + y^2\).
- The term \((x - y)^2\) involves a product of the dependent variable \(y\) and the independent variable \(x\) when multiplied by \(\frac{dy}{dx}\).
- This means that the dependent variable \(y\) and its derivative \(\frac{dy}{dx}\) are involved in a product.

**Conclusion:** This equation is not linear because it violates the condition of having no products of dependent variables and their derivatives.

### Final Answer
The differential equation that is not linear is the third one: \((x - y)^2 \cdot \frac{dy}{dx} = 9\).

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