Problem 30 - Calculus Concepts

Question

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. \(\frac{d y}{d x}=k x y\)

Step-by-Step Solution

Answer

The solution is found by separation of variables: \( y = C_1 e^{\frac{k x^2}{2}} \).

Step 1: Analyze the Type of Differential Equation

The given differential equation is \( \frac{d y}{d x} = k x y \). To identify the method of solution, recognize this equation as a \textbf{separable differential equation} because the variables can be separated into two distinct sides of the equation.

Step 2: Rearrange To Separate Variables

Separate the equation by dividing both sides by \( y \) and multiplying both sides by \( dx \):\[ \frac{1}{y} d y = k x d x \]

Step 3: Integrate Both Sides

Integrate both sides of the separated equation:\[ \int \frac{1}{y} d y = \int k x d x \]The left side integrates to \( \ln |y| \), and the right side integrates to \( \frac{k x^2}{2} + C \) where \( C \) is the constant of integration.

Step 4: Solve for y

Exponentiate both sides to solve for \( y \):\[ y = e^{\frac{k x^2}{2} + C} = e^C e^{\frac{k x^2}{2}} \]Letting \( e^C = C_1 \), we simplify to get:\[ y = C_1 e^{\frac{k x^2}{2}} \] where \( C_1 \) is an arbitrary constant.

Step 5: State the General Solution

The general solution to the differential equation \( \frac{d y}{d x} = k x y \) is:\[ y = C_1 e^{\frac{k x^2}{2}} \] where \( C_1 \) is an arbitrary constant.

Key Concepts

Separable Differential Equation

A separable differential equation is a type of differential equation in which the variables can be rearranged so that all terms involving one variable are on one side of the equation and all terms involving the other variable are on the other side. This allows for the equation to be solved through the process of integration.
In the original exercise, the differential equation given was \( \frac{d y}{d x} = k x y \). It is identified as separable because you can rewrite it by separating \( y \) and \( x \) into two distinct sides by rearranging the terms.
This transformation results in:

• Move all \( y \) terms to one side: \( \frac{1}{y} \ d y \)
• Move all \( x \) terms to the other side: \( k x \ d x \)

The act of separating these variables sets the stage for solving the equation through integration in the subsequent steps.

General Solution

Finding the general solution of a differential equation involves determining the most comprehensive form of the solution that can satisfy the equation across a range of possible cases.
The general solution includes one or more arbitrary constants, which can be adjusted based on initial or boundary conditions provided in specific problems.
In the provided step-by-step solution, the general solution is obtained after integration and simplification by solving for \( y \):

• Through integration, we reach \( \ln |y| = \frac{k x^2}{2} + C \)
• Solving for \( y \) gives \( y = C_1 e^{\frac{k x^2}{2}} \)

Here, \( C_1 = e^C \) is the arbitrary constant serving to fulfill the phrase "general solution" since it allows adjustments dependent on additional conditions.

Antiderivatives

An antiderivative, loosely speaking, is the reverse operation of a derivative. It recovers a function from its rate of change. In differential equations, particularly separable ones, antiderivatives play a key role during the integration process.
In this example, recall the step \( \int \frac{1}{y} \, d y = \int k x \, d x \).

• The left side, \( \int \frac{1}{y} \, d y \), integrates into \( \ln |y| \), which is the antiderivative of \( \frac{1}{y} \).
• Meanwhile, the right side, \( \int k x \, d x \), becomes \( \frac{k x^2}{2} \), the antiderivative of \( k x \).

Thus, antiderivatives convert the differential equation into a more easily solvable form.

Integration

Integration is the mathematical process of finding antiderivatives. In the context of solving differential equations, integration is critical in transitioning from rates of change back to the original function.
To integrate means to compute the area under the curve on a graph, so when integrating both sides of a separated differential equation, each integration effectively "undoes" the differentiation previously done to \( y \) and \( x \).
For the differential equation \( \frac{d y}{d x} = k x y \):

• We integrated \( \int \frac{1}{y} \, d y \) to get \( \ln |y| \)
• We integrated \( \int k x \, d x \) to result in \( \frac{k x^2}{2} \)

These integrals are where the magic happens and turns the separated equation into a form that can be solved for the function \( y \).

Calculus

Calculus is an essential branch of mathematics that deals with continuous change. It is composed of two primary operations: differentiation and integration.
Differential equations, a major application of calculus, describe a relationship between a function and its derivatives. They occur in naturally occurring phenomena and are used to model situations where rates of change are wanted.
In the problem presented, we utilize skills from calculus:

• Use differentiation knowledge to recognize the initial structure \( \frac{d y}{d x} \).
• Employ integration techniques to find the antiderivatives and solve for the general solution.

The power of calculus lets us solve complex real-world problems through this systematic approach.