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Chapter 11: Problem 28

Find the values of \(k\) for which \(y=x^{2}+k\) is a solution to the differential equation. $$ 2 y-x y^{\prime}=10 $$

Short Answer

Expert verified
The value of \(k\) is 5.

Step by step solution

01

Analyze the Given Solution

We are given the function \(y = x^2 + k\). To find the derivative \(y'\), differentiate with respect to \(x\).
02

Differentiate the Function

The derivative of \(y = x^2 + k\) with respect to \(x\) is \(y' = 2x\). This is because the derivative of \(x^2\) is \(2x\) and the derivative of a constant \(k\) is zero.
03

Substitute into the Differential Equation

Substitute \(y = x^2 + k\) and \(y' = 2x\) into the differential equation \(2y - xy' = 10\). This gives us:\[ 2(x^2 + k) - x(2x) = 10 \]
04

Simplify the Equation

Simplify the equation from Step 3:\[ 2x^2 + 2k - 2x^2 = 10 \]The \(2x^2\) terms cancel each other, simplifying to:\[ 2k = 10 \]
05

Solve for \(k\)

Solve the equation \(2k = 10\) by dividing both sides by 2:\[ k = \frac{10}{2} = 5 \]
06

Conclude the Solution

The solution to the differential equation using \(y = x^2 + k\) gives \(k = 5\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Derivative
The concept of a derivative is central to understanding calculus and the study of differential equations. In simple terms, a derivative represents the rate at which a function is changing at any given point. It's like the speed of a car at a particular moment, rather than its average speed over some distance. When we look at a function such as \( y = x^2 + k \), the derivative \( y' \) provides us with key information about the slope of the tangent line to the function at any point \( x \).
  • To get the derivative of \( y = x^2 + k \), we differentiate each term in the equation.
  • The derivative of \( x^2 \) is \( 2x \), because according to the power rule in calculus, \( x^n \)'s derivative is \( nx^{n-1} \).
  • The constant \( k \) vanishes upon differentiation because its rate of change with respect to \( x \) is zero.
Understanding derivatives is crucial as it allows us to explore the behavior of functions without graphing them explicitly.
Differentiation
Differentiation is the process of finding a derivative, and it's a fundamental operation in calculus. It is used to determine how a function changes, making it invaluable in fields ranging from physics to economics. Let's explore differentiation in the context of our function \( y = x^2 + k \).
  • To differentiate a function like \( y = x^2 + k \), employ rules such as the power rule and the constant rule.
  • The power rule tells us how to handle terms like \( x^2 \), producing the derivative \( 2x \).
  • The constant rule reminds us that the derivative of any constant is zero, as constants do not change with \( x \).
By differentiating correctly, we obtain \( y' = 2x \), which is needed for further investigation and to solve differential equations.
Mathematics Problem Solving
An essential skill in mathematics is being able to solve problems, and differential equations are excellent exercises for this. To solve a differential equation, follow a logical sequence of steps.First, you identify the given equation and its components. Here, the exercise provides the differential equation \( 2y - xy' = 10 \), and our candidate solution is \( y = x^2 + k \). Differentiation of \( y \) gives \( y' = 2x \).
  • Substitute \( y \) and \( y' \) into the differential equation to check for compatibility.
  • In this example, after simplification, the equation becomes \( 2k = 10 \), due to the cancellation of \( 2x^2 \) terms on either side.
  • Finally, solve for the variable \( k \) to find that \( k = 5 \).
Understanding the steps in this process is key, as it allows you to solve similar problems systematically. This method of problem-solving is widely applicable across different branches of science and engineering.

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Most popular questions from this chapter

Solve the boundary value problem. $$y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=1, y(1)=0$$

Find a solution to the following equation which satisfies \(z(0)=3\) and does not tend to infinity as \(t \rightarrow \infty\) $$ \frac{d^{2} z}{d t^{2}}+\frac{d z}{d t}-2 z=0 $$

Find the general solution to the given differential equation. $$z^{\prime \prime}+2 z=0$$

Find the general solution to the given differential equation. $$\frac{d^{2} x}{d t^{2}}+4 \frac{d x}{d t}+8 x=0$$

Give an example of: A system of differential equations for two populations \(X\) and \(Y\) such that \(Y\) needs \(X\) to survive and \(X\) is indifferent to \(Y\) and thrives on its own. Let \(x\) represent the size of the \(X\) population and \(y\) represent the size of the \(Y\) population.
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