Problem 22 - Calculus Early Transcendentals

Question

(a) Solve the equation \(y^{\prime}=\sqrt{y}\) and show that every nonconstant solution has a graph that is everywhere concave up. (b) Explain how the conclusion in part (a) may be obtained directly from the equation \(y^{\prime}=\sqrt{y}\) without solving.

Step-by-Step Solution

Answer

(a) Solve to get \( y = \left(\frac{x + C}{2}\right)^2 \); \( y'' = \frac{1}{2} > 0 \). (b) \( y'' = \frac{1}{2\sqrt{y}} \sqrt{y} > 0 \).

Step 1: Identify the differential equation

The given differential equation is \( y' = \sqrt{y} \). This is a separable differential equation, which indicates that we can separate the variables to solve it.

Step 2: Separate variables

To separate the variables, rewrite the equation as \( \frac{1}{\sqrt{y}} \, dy = dx \). This allows us to integrate both sides with respect to their respective variables.

Step 3: Integrate both sides

Integrate the left side \( \int \frac{1}{\sqrt{y}} \, dy \) which equals to \( 2 \sqrt{y} \). Integrate the right side \( \int dx \) which is simply \( x + C \), where \( C \) is a constant of integration. Thus, the equation becomes \( 2 \sqrt{y} = x + C \).

Step 4: Solve for \( y \)

To express \( y \) in terms of \( x \), rearrange the integration result to get \( \sqrt{y} = \frac{x + C}{2} \). Squaring both sides gives \( y = \left(\frac{x + C}{2}\right)^2 \).

Step 5: Find the second derivative

To show that the graph is everywhere concave up, compute the second derivative of \( y \). The first derivative was given \( y' = \sqrt{y} \). To find the second derivative, \( y'' = \frac{d}{dx}(y') \). Using \( y = \left(\frac{x + C}{2}\right)^2 \), differentiate to obtain \( y'' = \frac{1}{2} \).

Step 6: Conclude concavity of the solution

Since \( y'' = \frac{1}{2} > 0 \) for all \( x \), the graph of every nonconstant solution is concave up everywhere.

Step 7: Analyze without solving (part b)

We note from the original equation \( y' = \sqrt{y} \) that \( y' > 0 \) because \( \sqrt{y} > 0 \) for any positive \( y \). For the concavity, rearrange \( y'' \). Take a derivative with respect to \( x \) on both sides: \( y'' = \frac{d}{dx}(\sqrt{y}) = \frac{1}{2\sqrt{y}} y' = \frac{1}{2} \). Since this expression is positive for \( y > 0 \), it shows the graph is concave up throughout.

Key Concepts

Separable Differential Equations

Differential equations can often feel overwhelming. However, one of the most approachable types is a **separable differential equation**. As the name suggests, the key to solving these equations is to "separate" the variables. This means we can manipulate the equation so that each variable appears on a different side.

Here's how it works: consider the equation given as an example, \( y' = \sqrt{y} \). In a separable form, it can be rewritten as \( \frac{1}{\sqrt{y}} \, dy = dx \). Now, the terms involving \( y \) are all on one side, and \( x \) terms on the other.

• This separation allows us to integrate each side separately.
• For the given differential equation, integrating \( \int \frac{1}{\sqrt{y}} \, dy \) yields \( 2\sqrt{y} \), and integrating \( \int dx \) simply gives \( x + C \), where \( C \) is the integration constant.

The result is an expression that relates \( x \) and \( y \) explicitly, making it much easier to understand the behavior of the solution.

Concavity

One important aspect of any function's graph is its concavity. Concavity tells us how the graph bends or curves.

In mathematical terms, concavity is determined by the second derivative of a function.

• If the second derivative \( y'' \) is positive, the graph is concave up, resembling a "U" shape.
• If \( y'' \) is negative, the graph is concave down, similar to an upside down "U".

For the solved equation \( y = \left(\frac{x + C}{2}\right)^2 \), the second derivative is calculated as \( y'' = \frac{1}{2} \). Here, since \( \frac{1}{2} \) is a positive number, this indicates that the graph of the solution is concave up everywhere.

Catching this concept early on helps you predict the shape of graphs simply by examining the second derivative.

Nonconstant Solutions

A nonconstant solution is a solution that changes with respect to the independent variable—in this case, \( x \). Simply put, it isn't a flat line across the graph.

In our problem, the nonconstant solutions have a vital property: they're differentiable and offer dynamic graphs. Here's how you can identify and understand them:

• The function \( y = \left(\frac{x + C}{2}\right)^2 \) changes as \( x \) changes, making it nonconstant unless \( C \) somehow creates a constant.
• These solutions inherently follow the rules of differential calculus, allowing us to compute derivatives and analyze concavity for insights.

Understanding the distinction between constant and nonconstant solutions can significantly enhance your grasp on differential equations, allowing for a deeper comprehension of potential solutions and their graph's behavior.