Determine the degree of the differential equation `(1+((dy)/(dx))^(2))^((3)/(2)) = 5(d^(2)y)/(dx^(2))`. Also state the equation is linear or non-linear. Justify your answer.

Answer

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

**Step 1: Write down the given differential equation.**
The given differential equation is:
\[ (1 + \left(\frac{dy}{dx}\right)^2)^{\frac{3}{2}} = 5 \frac{d^2y}{dx^2} \]

**Step 2: Rearrange the equation to isolate the highest derivative.**
To analyze the degree, we need to express the equation in a form where we can identify the highest derivative clearly. We can start by squaring both sides to eliminate the exponent on the left side:
\[ \left(1 + \left(\frac{dy}{dx}\right)^2\right)^3 = 25 \left(\frac{d^2y}{dx^2}\right)^2 \]

**Step 3: Expand the left side using the binomial expansion.**
Using the binomial expansion for \((a + b)^3\), where \(a = 1\) and \(b = \left(\frac{dy}{dx}\right)^2\):
\[ 1 + 3\left(\frac{dy}{dx}\right)^2 + 3\left(\frac{dy}{dx}\right)^4 + \left(\frac{dy}{dx}\right)^6 = 25 \left(\frac{d^2y}{dx^2}\right)^2 \]

**Step 4: Identify the highest derivative and its power.**
Now, we can observe the equation:
\[ 1 + 3\left(\frac{dy}{dx}\right)^2 + 3\left(\frac{dy}{dx}\right)^4 + \left(\frac{dy}{dx}\right)^6 = 25 \left(\frac{d^2y}{dx^2}\right)^2 \]
The highest derivative present is \(\frac{d^2y}{dx^2}\), and its power is \(2\). Therefore, the degree of the differential equation is \(2\).

**Step 5: Determine if the equation is linear or non-linear.**
To classify the equation, we check if it can be expressed in the form of a linear combination of the derivatives of \(y\). A linear differential equation can be written as:
\[ a_n(x) \frac{d^n y}{dx^n} + a_{n-1}(x) \frac{d^{n-1} y}{dx^{n-1}} + \ldots + a_1(x) \frac{dy}{dx} + a_0(x)y = g(x) \]
In our case, the presence of terms like \(\left(\frac{dy}{dx}\right)^4\) and \(\left(\frac{dy}{dx}\right)^6\) indicates that the equation is not linear because it contains powers of the derivatives greater than one.

**Final Answer:**
- The degree of the differential equation is \(2\).
- The equation is non-linear.