Problem 11 - Applied Calculus

Question

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ \begin{aligned} &y^{\prime}=\frac{y}{x}\\\ &\text { and check } \end{aligned} $$

Step-by-Step Solution

Answer

The general solution is \( y = Cx \), where \( C \) is a constant.

Step 1: Identify Type of Differential Equation

The differential equation given is \( y' = \frac{y}{x} \). This can be recognized as a separable differential equation because we can express it in the form \( \frac{dy}{dx} = g(x)h(y) \), where \( g(x) = \frac{1}{x} \) and \( h(y) = y \).

Step 2: Separate Variables

Separate the variables by multiplying both sides by \( x \) and \( dx \) to get \( y' \cdot dx = \frac{y}{x} \cdot dx \rightarrow y \cdot dx = x \cdot dy \). Rearranging gives \( \frac{dy}{y} = \frac{dx}{x} \).

Step 3: Integrate Both Sides

Integrate both sides of the equation: \( \int \frac{dy}{y} = \int \frac{dx}{x} \). The integrals yield \( \ln |y| = \ln |x| + C \), where \( C \) is the constant of integration.

Step 4: Solve for \( y \)

To solve for \( y \), exponentiate both sides to remove the logarithms: \( e^{\ln |y|} = e^{\ln |x| + C} \). This simplifies to \( |y| = e^C |x| \), or \( y = \pm e^C x \). Letting \( e^C = C' \), the solution becomes \( y = C'x \).

Step 5: Check Solution

To verify, substitute \( y = C'x \) into the original equation \( y' = \frac{y}{x} \). We first compute \( y' = \frac{d}{dx}(C'x) = C' \). Substituting into the original equation gives \( C' = \frac{C'x}{x} = C' \), confirming that \( y = C'x \) is indeed a solution.

Key Concepts

Separable Differential Equations

Separable differential equations are a special class of differential equations that can be broken down into the separate functions of two independent variables. These equations are particularly user-friendly because they allow both sides to be integrated independently.
For a differential equation to be separable, it should be possible to rearrange the equation into the form \( \frac{dy}{dx} = g(x)h(y) \). This means that you can express it as a product of a function of \( x \) and a function of \( y \). Essentially, you want to move all terms involving \( y \) to one side and all terms involving \( x \) to the other side.
In the problem we have, \( y' = \frac{y}{x} \) is indeed separable. By recognizing that we can rewrite this equation as \( \frac{dy}{dx} = \frac{y}{x} \), we see that it fits the pattern, which then allows us to proceed to the next steps of solving it.

Integration Techniques

In the context of separable differential equations, integration techniques play a crucial role in finding a solution. After separating the variables, the next step involves integrating each side of the equation.
For our problem, after separation, we have \( \frac{dy}{y} = \frac{dx}{x} \). This is where the power of integration comes into play. Integrating the left-hand side, \( \int \frac{dy}{y} \), results in the natural logarithm, \( \ln |y| \). Similarly, integrating \( \int \frac{dx}{x} \) on the right-hand side gives us \( \ln |x| + C \), where \( C \) is the constant of integration.
Only through integration can we transition from a differential equation to an equation involving no derivatives, thereby getting one step closer to an explicit solution for \( y \). This illustrates why mastering integration techniques is fundamental for solving differential equations.

General Solution

A general solution of a differential equation is a solution that includes an arbitrary constant. This constant represents an infinite number of possible solutions.
In solving separable differential equations like \( y' = \frac{y}{x} \), after integration, we arrive at \( \ln |y| = \ln |x| + C \). The next step is to solve for \( y \), which involves removing the logarithmic function by exponentiation.
Exponentiating both sides, we have \( e^{\ln |y|} = e^{\ln |x| + C} \). This simplifies to \( |y| = e^{C} |x| \), hence \( y = \pm e^{C} x \). To simplify, we often represent \( e^{C} \) by another constant, say \( C' \), leading to a general solution of the form \( y = C'x \).
This form of solution provides immense flexibility because varying \( C' \) gives us a family of functions, all of which satisfy the original differential equation. Thus, the concept of a general solution is at the heart of understanding differential equations and represents a crucial step in their analysis.