Question

For the following problems, find the general solution to the differential equation.\(y^{\prime}=\frac{y}{x}\)

Short answer

The general solution is \( y = C_1 x \).

Step-by-step solution

Identify the Type of Differential Equation

The given differential equation is \( y' = \frac{y}{x} \). This is a first-order differential equation and can be categorized as a separable equation because the variables \( y \) and \( x \) can be separated.

Separate the Variables

Rearrange the equation to separate the variables \( y \) and \( x \) on different sides of the equation:\[ \frac{dy}{y} = \frac{dx}{x} \]

Integrate Both Sides

Integrate both sides of the separated equation:\[ \int \frac{dy}{y} = \int \frac{dx}{x} \]

Solve the Integrals

The integral of \( \frac{dy}{y} \) is \( \ln |y| \) and the integral of \( \frac{dx}{x} \) is \( \ln |x| \). Therefore:\[ \ln |y| = \ln |x| + C \]where \( C \) is the constant of integration.

Eliminate Logarithms and Solve for y

Exponentiate both sides to eliminate the logarithms:\[ |y| = e^{\ln |x| + C} \]which simplifies to:\[ y = \pm e^C x \]Note that \( e^C \) can be replaced by a new constant \( C_1 \). So: \[ y = C_1 x \], where \( C_1 = \pm e^C \).

Write the General Solution

The general solution of the differential equation \( y' = \frac{y}{x} \) is:\[ y = C_1 x \]where \( C_1 \) is an arbitrary constant.

Key concepts

First-order Differential Equation

A first-order differential equation involves the first derivative of a function and no higher derivatives. In the equation provided, \( y' = \frac{y}{x} \), the expression \( y' \) signifies the first derivative of \( y \) with respect to \( x \).
This type of equation is frequently seen in various fields like physics and engineering, where it often describes rates of change.
The goal is to determine the function \( y(x) \) that satisfies the equation.Key points about first-order differential equations include:

• Involving only the first derivative and no higher order derivatives.
• Requiring initial conditions for specific solutions.
• Often solved by methods like separation of variables, substitution, or integrating factors.

Understanding the nature of first-order differential equations is crucial, as they form the base for understanding more complex differential equations.

Variable Separation

Variable separation is a powerful technique for solving differential equations where we can rearrange the equation to isolate different variables on each side. This strategy simplifies the equation into a form we can integrate easily.
For example, consider the equation \( y' = \frac{y}{x} \):

• Separate \( y \) and \( x \) such that \( \frac{dy}{y} = \frac{dx}{x} \).
• This approach effectively isolates the variables making it ready for integration.

The reason why variable separation is particularly useful is that it simplifies the process of solving complex-looking differential equations by breaking them down into manageable parts.
It ensures each part involves only one variable, allowing for straightforward application of integration techniques.

Integration Techniques

Once the variables are separated in a differential equation, the next step is to integrate both sides. This involves using basic integration techniques to find the antiderivative. In our example, we have:

• \( \int \frac{dy}{y} = \ln |y| \), a common integral known from basic calculus as the natural logarithm of the absolute value of \( y \).
• \( \int \frac{dx}{x} = \ln |x| \), similarly resulting in the natural logarithm of the absolute value of \( x \).

After integration, the equation often includes a constant of integration \( C \), which represents the family of solutions.
In the step-by-step solution, exponentiation is used to eliminate logarithms, leading to \( |y| = e^{C}x \). This can be simplified further to the general solution, \( y = C_1 x \), where \( C_1 \) is an arbitrary constant.
Recognizing when to apply these basic integration techniques allows one to solve a host of first-order separable differential equations efficiently.