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Chapter 1: Problem 11

Verify that the given function is a solution to the given differential equation \(\left(c_{1} \text { and } c_{2}\right.\) are arbitrary constants), and state the maximum interval over which the solution is valid. $$y(x)=c_{1} x^{1 / 2}, \quad y^{\prime}=\frac{y}{2 x}$$.

Short Answer

Expert verified
The given function \(y(x) = c_1x^{1/2}\) is a solution to the differential equation \(y' = \frac{y}{2x}\), as its derivative \(y'(x) = \frac{1}{2}c_1x^{-1/2}\) satisfies the equation. The maximum interval for the solution is \(x > 0\) or in interval notation, \( (0, \infty)\).

Step by step solution

01

Find the derivative of the given function

First, let's find the derivative of the given function \(y(x) = c_1x^{1/2}\) with respect to \(x\). Applying the power rule of differentiation, we get: \[y'(x) = \frac{1}{2}c_1x^{-1/2}\]
02

Substitute the derivatives in the differential equation

Now let's substitute our obtained values of \(y(x)\) and \(y'(x)\) into the given differential equation \(y' = \frac{y}{2x}\). \[\frac{1}{2}c_1x^{-1/2} = \frac{c_1x^{1/2}}{2x}\]
03

Simplify the equation and verify the function

Simplify the right-hand side of the equation: \[\frac{1}{2}c_1x^{-1/2} = \frac{c_1x^{-1/2}}{2}\] The equation holds, since both sides of the equation are equal. Hence, we can conclude that the given function \(y(x) = c_1x^{1/2}\) is indeed a solution to the given differential equation \(y' = \frac{y}{2x}\).
04

Determine the maximum interval for the solution

The given function \(y(x) = c_1x^{1/2}\) is defined for all real positive values of \(x\), as the square root is undefined for negative values of \(x\). Therefore, the maximum interval over which the solution is valid is: \(x > 0\) or in interval notation, \( (0, \infty)\).

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Key Concepts

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

Power Rule of Differentiation
The power rule of differentiation is a fundamental tool in calculus, especially when it comes to solving differential equations.
When dealing with functions like \(y = x^n\), where \(n\) is any real number, finding the derivative is straightforward using this rule.
To apply the power rule, simply multiply the power \(n\) by the coefficient of \(x\), and then subtract one from the exponent. The general formula is: \[ \frac{d}{dx}(x^n) = nx^{n-1} \]
For example, if \(y(x) = c_1x^{1/2}\), using the power rule gives us the derivative \(y'(x)\) as \(\frac{1}{2}c_1x^{-1/2}\).
Derivative of a Function
Taking the derivative of a function means finding the rate at which the function's value changes at any given point.
It's like zooming in on a point on a curve to see how steep it is.
Derivatives are essential in understanding the behavior of functions, as they give insight into the growth or decay rate.
In our exercise, after applying the power rule, we find that the derivative of \(y(x) = c_1x^{1/2}\) is \(y'(x) = \frac{1}{2}c_1x^{-1/2}\), which is crucial for verifying the solution to the differential equation.
Valid Interval for Solution
The valid interval for a solution to a differential equation is the range of the independent variable, often \(x\), for which the solution is defined and meets the requirements of the equation.
Not all functions behave nicely across all values of \(x\). They might become undefined or reach values that don鈥檛 fit into the real numbers.
For instance, in the solved exercise, the function \(y(x) = c_1x^{1/2}\) only makes sense for positive values of \(x\), because you cannot take the square root of a negative number without leaving the realm of real numbers.
Therefore, the solution is valid for \(x > 0\), or in interval notation, \((0, \frac{1}{2}]\).

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Most popular questions from this chapter

Determine which of the five types of differential equations we have studied the given equation falls into (see Table \(1.12 .1),\) and use an appropriate technique to find the general solution. $$\left(1+2 x e^{y}\right) d x-\left(e^{y}+x\right) d y=0$$

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. Cobalt-60, an isotope used in cancer therapy, decays exponentially with a half-life of 5.2 years (i.e., half the original sample remains after 5.2 years). How long does it take for a sample of Cobalt-60 to disintegrate to the extent that only \(4 \%\) of the original amount remains?

Solve $$ \sec ^{2} y \frac{d y}{d x}+\frac{1}{2 \sqrt{1+x}} \tan y=\frac{1}{2 \sqrt{1+x}} $$

Use the result from the previous problem to solve the given differential equation. For Problem 54 impose the given initial condition as well. $$y^{\prime}=(9 x-y)^{2}, \quad y(0)=0$$

Consider the phenomenon of exponential decay. This occurs when a population \(P(t)\) is governed by the differential equation $$\frac{d P}{d t}=k P$$ where \(k\) is a negative constant. At the conclusion of the Super Bowl, the number of fans remaining in the stadium decreases at a rate proportional to the number of fans in the stadium. Assume that there are 100,000 fans in the stadium at the end of the Super Bowl and ten minutes later there are 80,000 fans in the stadium. (a) Thirty minutes after the Super Bowl will there be more or less than 40,000 fans? How do you know this without doing any calculations? (b) What is the half-life (see the previous problem) for the fan population in the stadium? (c) When will there be only 15,000 fans left in the stadium? (d) Explain why the exponential decay model for the population of fans in the stadium is not realistic from a qualitative perspective.
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