Problem 11 - Differential Equations and Linear Algebra

Question

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. $$y(x)=c_{1} x^{1 / 2}, \quad y^{\prime}=\frac{y}{2 x}$$.

Step-by-Step Solution

Answer

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 1: 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: \[y'(x) = \frac{1}{2}c_1x^{-1/2}\]

Step 2: 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 \(y' = \frac{y}{2x}\). \[\frac{1}{2}c_1x^{-1/2} = \frac{c_1x^{1/2}}{2x}\]

Step 3: 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 \(y' = \frac{y}{2x}\).

Step 4: 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, \( (0, \infty)\).

Key Concepts

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’t 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, \((0, \frac{1}{2}]\).