Question

For what value of the exponent a is the function \(\mathrm{y}=\mathrm{x}^{\mathrm{a}}\) a solution to the differential equation\(\frac{\mathrm{dy}}{\mathrm{dx}}=-\mathrm{y}^{2} ?\)

Short answer

Question: Find the value of the exponent a for which the function y=x^a is a solution to the differential equation dy/dx = -y^2. Answer: The value of the exponent a is 1/3.

Step-by-step solution

Find the derivative of y=x^a

To find the derivative of the function y=x^a, we can use the power rule, which states that the derivative of x^n is nx^(n-1). So, the derivative of x^a is: dy/dx = a * x^(a-1)

Substitute y=x^a and its derivative into the differential equation

We are given the differential equation dy/dx = -y^2. We can substitute the expression for y (x^a) and the expression for dy/dx (a * x^(a-1)) into this equation: a * x^(a-1) = -(x^a)^2

Simplify the equation

By simplifying the right side of the equation, we get: a * x^(a-1) = -x^(2a) At this point we have to find a way to equate the exponents of x on both sides, so: a-1 = 2a Given that x ≠ 0, this equation must be true for the given function to be a solution to the differential equation.

Solve for a

By solving the equation a-1 = 2a, we obtain the value for the exponent a: 1 = 3a a = 1/3

Conclusion

The value of the exponent a for which the function y=x^a is a solution to the differential equation dy/dx = -y^2 is a = 1/3.

Key concepts

Power Rule Differentiation

Power rule differentiation is a fundamental technique in calculus for finding the derivative of a function in the form of y = x^n, where n is any real number. The power rule states that the derivative of x^n is nx^(n-1). In simpler terms, you take the exponent n, multiply the function by it, and then subtract one from the exponent.

For example, if you have a function f(x) = x^3, applying the power rule gives us f'(x) = 3x^(3-1), which simplifies to f'(x) = 3x^2. The power rule is often used because of its simplicity and ability to easily handle functions with polynomial-like terms. It is a key tool for solving a wide variety of problems in calculus, including differential equations, where finding the derivative is a critical first step.

Solving Differential Equations

A differential equation is an equation that involves a function and its derivatives. Solving differential equations means finding a function or set of functions that satisfy the given equation. They come in many varieties, including ordinary differential equations (ODEs) and partial differential equations (PDEs), and can be linear or nonlinear.

Differential equations are essential in math and physics as they describe various phenomena such as motion, heat flow, seismic waves, and population dynamics. The process of solving them typically involves finding the derivatives using rules like the power rule and then integrating or isolating variables to solve for the unknown function. For instance, in the context of the exercise provided, the differential equation dy/dx = -y^2 is solved by substituting the expression for y and its derivative, and then finding the exponent value that satisfies the equation.

Exponent Value

In the context of solving differential equations, the exponent value often plays a crucial role in determining the solution of the function. The exponent refers to the power to which a number is raised. When a function such as y = x^a is part of a differential equation, finding the correct exponent value a is essential for the function to satisfy the equation.

In the exercise example, we are tasked with finding the value of a such that x^a solves a given differential equation. By using the power rule for differentiation, balancing the exponents, and then solving for a, as shown in the step-by-step solution, we discover that the exponent value which makes the function an appropriate solution is a = 1/3. This reflects not just the mathematical manipulation but also the importance of exponent values in defining the behavior of functions within differential equations.

Integral Calculus

Integral calculus, often paired with differential calculus, is concerned with finding the quantity where the rate of change (the derivative) is known. Its primary operation is integration, the inverse process of differentiation. While differentiation converts a function into its rate of change, integration takes a rate of change and reconstructs the original function.

Integrals can be used to calculate areas, volumes, central points, and many other useful things. But in the realm of differential equations, integral calculus is used to find the unknown function that the differential equation describes. After finding the derivative using rules like the power rule, the resulting expression (which might be an equation or an inequality) can often be integrated to find a solution to the original problem.