Let's first break down the question and its components:

Definitions:

1. Linear Differential Equation: A differential equation is considered linear if the dependent variable $y$ and its derivatives appear to the first power and are not multiplied or divided by each other.

2. Non-Linear Differential Equation: A differential equation is non-linear if any term contains powers, products, or other non-linear operations (like square roots or trigonometric functions) involving the dependent variable $y$ or its derivatives.

3. Order of a Differential Equation: The order of a differential equation is the highest derivative present in the equation.

4. Degree of a Differential Equation: The degree of a differential equation is the power of the highest derivative after the equation is made free from radicals and fractions involving derivatives.

Now, let's address the specific differential equation from the image:

$\sqrt[4]{\left(\frac{d^3 y}{dx^3}\right)^4} - 5x \cdot \frac{d^2 y}{dx^2} + y = 5 \sqrt{\left(\frac{dy}{dx}\right)^2} + y^2 - x$

Step-by-step solution:

1. Simplify the equation:
   • The term $\sqrt[4]{\left(\frac{d^3 y}{dx^3}\right)^4}$ simplifies to $\frac{d^3 y}{dx^3}$, because taking the fourth root of the fourth power cancels out.
   • The term $5 \sqrt{\left(\frac{dy}{dx}\right)^2}$ simplifies to $5 \left|\frac{dy}{dx}\right|$, which is the absolute value.

So, the equation becomes:

$\frac{d^3 y}{dx^3} - 5x \cdot \frac{d^2 y}{dx^2} + y = 5 \left|\frac{dy}{dx}\right| + y^2 - x$

2. Identify if the equation is linear or non-linear:
   • The presence of $\left|\frac{dy}{dx}\right|$ and $y^2$ makes the equation non-linear because these terms are non-linear in the dependent variable $y$ and its derivatives.

Thus, the equation is non-linear.

3. Order of the equation:

   • The highest derivative present is $\frac{d^3 y}{dx^3}$, which makes the order of the differential equation 3.

4. Degree of the equation:

   • To find the degree, we first eliminate any radicals or absolute value signs, but since the term $\left|\frac{dy}{dx}\right|$ complicates this, it's not possible to assign a degree based on the standard approach.
   • As a result, this equation does not have a well-defined degree in its current form.

Final Answer:

• Type: Non-linear
• Order: 3
• Degree: Not well-defined due to absolute value and non-linear terms.

Would you like further details or clarifications on any specific part of this?