Problem 36 - Calculus

Question

Write down a differential equation \(d y / d x=\) _______ that is solved by \(y=x^{2}\). Make the right side involve \(y\) (not just \(2 x\) ).

Step-by-Step Solution

Answer

\(\frac{dy}{dx} = 2\sqrt{y}\)

Step 1: Differentiate the given function

The given function is \( y = x^2 \). To find the differential equation that has this form as its solution, we first differentiate this function with respect to \( x \). The derivative of \( y = x^2 \) is \( \frac{dy}{dx} = 2x \).

Step 2: Express \(x\) in terms of \(y\)

Since the problem asks for the differential equation to involve \( y \), not just \( x \), we need to substitute \( x \) in terms of \( y \). From the equation \( y = x^2 \), we can express \( x \) in terms of \( y \): \( x = \sqrt{y} \).

Step 3: Substitute back into the derivative

Substitute \( x = \sqrt{y} \) back into the derivative \( \frac{dy}{dx} = 2x \). This gives us \( \frac{dy}{dx} = 2 \sqrt{y} \).

Step 4: Formulate the differential equation

The differential equation that has \( y = x^2 \) as a solution, expressed in terms of \( y \), is \( \frac{dy}{dx} = 2 \sqrt{y} \).

Key Concepts

Derivative

A derivative represents how a function changes as its input changes; it's the fundamental idea behind calculus. In simpler terms, it measures the rate at which one quantity changes compared to another. For example, if you are looking at a function that describes the position of a car over time, the derivative of this function with respect to time tells you the car's speed.

• **Notation:** The derivative of a function \( y \) with respect to \( x \) is usually denoted as \( \frac{dy}{dx} \).
• **Meaning:** It gives us the slope of the tangent line to the curve at any point, which is a measure of how steep the curve is at that point.
• **Calculation:** For a simple function like \( y = x^2 \), the derivative is obtained through rules of differentiation, such as the power rule.

In the exercise, differentiating \( y = x^2 \) gives \( \frac{dy}{dx} = 2x \). This tells us that for any point on this parabola, the slope of the tangent line is proportional to the x-coordinate, doubled.

Solution of a Differential Equation

Finding a solution to a differential equation means finding a function or a set of functions that satisfy the equation under given conditions. In this exercise, we start with the solution, \( y = x^2 \), and work backwards to find a differential equation it satisfies.

• **Expression:** A solution to a differential equation is a function that makes the differentiation equation true.
• **Verification:** After finding or guessing a potential solution, substituting it back into the original differential equation verifies its validity.
• **General and Particular Solutions:** Solutions can be general (having arbitrary constants) or particular (specific values making it unique).

In our exercise, \( y = x^2 \) was confirmed as a valid solution by deriving \( \frac{dy}{dx} = 2 \sqrt{y} \), showing that it satisfies the derived differential equation when plugged back.

Substitution Method

The substitution method is a powerful tool to simplify and resolve differential equations or expressions that might seem complex at first. In many cases, substitution makes the problem easier to solve by transforming the original equation into a more recognizable or manageable form.

• **Purpose:** It helps to eliminate one of the variables or express one variable in terms of others, aiding in simplifying the equation.
• **Method:** Identify an expression you can replace, substitute it with a new variable or expression to simplify the problem.
• **Example from Exercise:** In the step-by-step solution, we substituted \( x = \sqrt{y} \) into \( \frac{dy}{dx} = 2x \), transforming it into \( \frac{dy}{dx} = 2 \sqrt{y} \).

This allowed the problem to be expressed entirely in terms of \( y \), which was required by the problem prompt. Like here, many complex problems can be made simpler through strategic substitution.