Understanding and Deriving Differential Equations from Given Functions

This article aims to guide you through the process of deriving differential equations from given primitive functions. Specifically, we will explore the primitive function y Ax^{2} - Bx and derive the corresponding differential equation. This process involves differentiation, constant elimination, and forming a relationship between the function and its derivatives.

Deriving the Differential Equation for y Ax^{2} - Bx

To begin, we will differentiate the given primitive function y Ax^{2} - Bx with respect to x to find the first derivative.

First Derivative

Starting with:

y Ax^{2} - Bx

We differentiate both sides with respect to x to find the first derivative:

(frac{dy}{dx} 2Ax - B)

This expression gives us a first-order differential equation relating y and its first derivative (frac{dy}{dx}).

Eliminating Constants

To eliminate the constants A and B, we can express them in terms of y and x.

From the original equation:

A frac{y - Bx}{x^{2}})

And from the first derivative:

B frac{dy}{dx} - 2Ax)

Eliminating these constants can be complex, so a more straightforward approach is to form a second-order differential equation by differentiating the first derivative once more.

Second Derivative

Re-differentiate (frac{dy}{dx}) with respect to x to find the second derivative:

(frac{d^{2}y}{dx^{2}} 2A)

Since A is a constant, the second derivative is constant and can be denoted as k (i.e., (frac{d^{2}y}{dx^{2}} k)).

Therefore, the differential equation corresponding to the primitive y Ax^{2} - Bx is (frac{d^{2}y}{dx^{2}} k).

Additional Example: y Ax^{3} - Bx^{2}

Let's consider the case where we have a primitive function y Ax^{3} - Bx^{2}. The process is similar but more complex due to the higher degree terms.

Solving for A and B

Expressing A in terms of y and x:

A frac{y - Bx^{2}}{x^{3}})

Differentiate with respect to x to find the first derivative:

(y' 3Ax^{2} - 2Bx)

Expressing B in terms of y' and x:

B frac{y' - 2Ax}{2x})

Substituting and simplifying, we can derive the differential equation:

(frac{d^{2}y}{dx^{2}} - 4frac{dy}{dx} 6y 0)

This is an Euler-Cauchy differential equation with a generic solution of the form y k_{1}x^{3} k_{2}x^{2}.

Simplifying the Problem: y Cx^{2}

Another approach is to simplify the problem by starting with a simpler primitive function y Cx^{2}. This simplification can help us derive a differential equation without the complexities involved in the previous functions.

Deriving the Differential Equation

Starting with the simplified function:

y Cx^{2}

First, differentiate with respect to x:

(frac{dy}{dx} 2Cx)

Now, differentiate the first derivative with respect to x:

(frac{d^{2}y}{dx^{2}} 2C)

Since C is a constant, we denote it as k, giving us:

(frac{d^{2}y}{dx^{2}} k)

This is the differential equation for the function y Cx^{2}.

Conclusion

Through this exploration, we have seen different methods to derive differential equations from given primitive functions. Whether through direct differentiation, constant elimination, or simplification, these methods can help us understand and solve complex mathematical problems in calculus.

Key Takeaways:

Differential Equation: A mathematical equation that relates a function with its derivatives. Calculus: Mathematical branch that deals with rates of change and slopes of curves. Primitive Function: A function whose derivative is the given function.