Determine the order and degree or each of the following. Also, state whether they are linear or non-linear:
`(d^(2)y)/(dx^(2))=cos3x+sin 3x`.

Answer

To determine the order and degree of the given differential equation, as well as to classify it as linear or non-linear, we can follow these steps:

### Step 1: Identify the given differential equation
The given differential equation is:
\[ \frac{d^2y}{dx^2} = \cos(3x) + \sin(3x) \]

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

**Order = 2**

### Step 3: 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. Here, the highest derivative \(\frac{d^2y}{dx^2}\) appears to the first power (i.e., it is not raised to any power other than 1).

**Degree = 1**

### Step 4: Determine if the differential equation is linear or non-linear
A differential equation is considered linear if it can be expressed in the form:
\[ a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x)y = g(x) \]
where \(a_n, a_{n-1}, \ldots, a_0\) are functions of \(x\) and \(g(x)\) is a function of \(x\) (not involving \(y\) or its derivatives).

In our equation:
\[ \frac{d^2y}{dx^2} = \cos(3x) + \sin(3x) \]
can be rearranged to:
\[ \frac{d^2y}{dx^2} - \cos(3x) - \sin(3x) = 0 \]
This form shows that it is linear in terms of \(y\) and its derivatives.

**Classification: Linear**

### Final Summary
- **Order:** 2
- **Degree:** 1
- **Type:** Linear differential equation

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