Find the complete primitive and singular soln(if exist)of y=2px+p⁴x²

To find the complete primitive and singular solution of the equation \( y = 2px + p^4x^2 \), we first need to clarify what we mean by "primitive" and "singular" solutions in this context. A primitive solution typically refers to the antiderivative of a function, while a singular solution often pertains to a specific solution that cannot be derived from the general solution of a differential equation. Let's break this down step by step.

Understanding the Function

The given function is \( y = 2px + p^4x^2 \). Here, \( p \) is treated as a parameter. This function is a polynomial in \( x \), and we can analyze it to find its antiderivative.

Finding the Primitive Solution

To find the primitive (or antiderivative) of the function with respect to \( x \), we will integrate the expression:

• First, rewrite the function: \( y = 2px + p^4x^2 \).
• Now, integrate term by term:

The integral of \( 2px \) with respect to \( x \) is:

\( \int 2px \, dx = px^2 + C_1 \)

where \( C_1 \) is a constant of integration. Next, we integrate \( p^4x^2 \):

\( \int p^4x^2 \, dx = \frac{p^4}{3}x^3 + C_2 \)

Combining these results, we have:

\( Y = px^2 + \frac{p^4}{3}x^3 + C \)

where \( C \) is a constant that combines \( C_1 \) and \( C_2 \).

Identifying Singular Solutions

Now, let’s consider singular solutions. A singular solution typically arises in the context of differential equations, particularly when the general solution does not encompass all possible behaviors of the system. In this case, we can analyze the function for specific values of \( p \).

If we set \( p = 0 \), the function simplifies to:

\( y = 0 \)

This is a trivial solution, but it is indeed a singular solution since it does not arise from the general form when \( p \) is non-zero. For other values of \( p \), the function remains a polynomial and does not yield any additional singular solutions.

Summary of Findings

In summary, the complete primitive solution of the function \( y = 2px + p^4x^2 \) is:

\( Y = px^2 + \frac{p^4}{3}x^3 + C \)

And the singular solution occurs when \( p = 0 \), leading to:

\( y = 0 \)

This analysis illustrates how we can derive both the primitive and singular solutions from the given polynomial function. If you have further questions or need clarification on any part of this process, feel free to ask!