Problem 20
Question
Is there a value of \(n\) which makes \(y=x^{n}\) a solution to the equation \(13 x(d y / d x)=y ?\) II so, what value?
Step-by-Step Solution
Verified Answer
The value of \( n \) is \( \frac{1}{13} \).
1Step 1: Understand the Problem
We need to check if there exists a value of \( n \) such that \( y = x^n \) is a solution to the given differential equation \( 13x \frac{dy}{dx} = y \). To find this, we'll differentiate \( y = x^n \) and substitute into the equation.
2Step 2: Differentiate the Function
Differentiate \( y = x^n \) with respect to \( x \). Using the power rule, we get: \( \frac{dy}{dx} = nx^{n-1} \).
3Step 3: Substitute into the Differential Equation
Substitute \( y = x^n \) and \( \frac{dy}{dx} = nx^{n-1} \) into the differential equation: \[ 13x(nx^{n-1}) = x^n \]. Simplify this equation to get \[ 13nx^n = x^n \].
4Step 4: Simplify the Equation
To find \( n \), divide both sides of the equation \( 13nx^n = x^n \) by \( x^n \), yielding \( 13n = 1 \).
5Step 5: Solve for \( n \)
Solve the equation \( 13n = 1 \) for \( n \) by dividing both sides by 13, resulting in \( n = \frac{1}{13} \).
Key Concepts
Power RuleDifferentiationDifferential Equation Solution
Power Rule
The power rule is a fundamental tool in calculus that simplifies the differentiation of expressions in which a variable is raised to a particular power. If you have a function of the form \( y = x^n \), where \( n \) is any real number, the power rule allows you to find its derivative with ease.
The rule states that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \). This means you multiply the original power by the coefficient and then subtract one from the original power.
This tool is especially handy when dealing with polynomials, as it allows quick determination of the rate of change at any given point. Using the power rule simplifies computations in calculus and is a key step in solving differential equations related to polynomials.
The rule states that the derivative of \( x^n \) with respect to \( x \) is \( nx^{n-1} \). This means you multiply the original power by the coefficient and then subtract one from the original power.
This tool is especially handy when dealing with polynomials, as it allows quick determination of the rate of change at any given point. Using the power rule simplifies computations in calculus and is a key step in solving differential equations related to polynomials.
Differentiation
Differentiation is a central concept in calculus and involves finding the derivative of a function.
The derivative represents the rate at which a function changes at any point in its domain, often interpreted as the slope of the tangent line to the function's graph at a particular point. In the context of the exercise, differentiation helps determine whether the function \( y = x^n \) satisfies the given differential equation.
When you differentiate \( y = x^n \) using the power rule, you get \( \frac{dy}{dx} = nx^{n-1} \). Integrating this into other calculus operations like solving differential equations helps us understand how functions react to changes in input, which is essential in areas such as physics, engineering, and economics.
The derivative represents the rate at which a function changes at any point in its domain, often interpreted as the slope of the tangent line to the function's graph at a particular point. In the context of the exercise, differentiation helps determine whether the function \( y = x^n \) satisfies the given differential equation.
When you differentiate \( y = x^n \) using the power rule, you get \( \frac{dy}{dx} = nx^{n-1} \). Integrating this into other calculus operations like solving differential equations helps us understand how functions react to changes in input, which is essential in areas such as physics, engineering, and economics.
Differential Equation Solution
Solving a differential equation involves finding a function that satisfies the equation. In the exercise, we are tasked with discovering if \( y = x^n \) is a solution to the equation \( 13x \frac{dy}{dx} = y \).
This involves substituting the derivative and the function itself back into the equation and checking for consistency.
Here, after finding \( \frac{dy}{dx} = nx^{n-1} \) and substituting it, the equation simplifies to \( 13nx^n = x^n \).
This involves substituting the derivative and the function itself back into the equation and checking for consistency.
Here, after finding \( \frac{dy}{dx} = nx^{n-1} \) and substituting it, the equation simplifies to \( 13nx^n = x^n \).
- Dividing both sides by \( x^n \) solves the equation for \( n \).
- The solution \( n = \frac{1}{13} \) indicates the specific value of \( n \) that makes \( y = x^n \) satisfy the differential equation.
Other exercises in this chapter
Problem 19
(a) What are the equilibrium solutions for the differential equation $$\frac{d y}{d t}=0.2(y-3)(y+2) ?$$ (b) Use a graphing calculator or computer to sketch a s
View solution Problem 19
Suppose \(Q=C e^{k t}\) satisfies the differential equation $$\frac{d Q}{d t}=-0.03 Q$$ What (if anything) does this tell you about the values of \(C\) and \(k
View solution Problem 21
A yam is put in a \(200^{\circ} \mathrm{C}\) oven and heats up according to the differential equation \(\frac{d H}{d t}=-k(H-200), \quad\) for \(k\) a positive
View solution Problem 21
Find the values of \(k\) for which \(y=x^{2}+k\) is a solution to the differential equation \(2 y-x y^{\prime}=10\)
View solution