Problem 35 - Calculus and its applications

Question

Solve by separating variables. \(\frac{d y}{d x}=4 x^{3} y\)

Step-by-Step Solution

Answer

The solution is: \( y = Ce^{x^4} \), where \( C \) is a constant.

Step 1: Separate the Variables

To solve the differential equation by separating variables, we first separate the terms involving \(dy\) and \(y\) on one side and \(dx\) and \(x\) on the other side. Starting from \( \frac{dy}{dx} = 4x^3y \), we can rewrite it as \( \frac{dy}{y} = 4x^3 dx \).

Step 2: Integrate Both Sides

Next, we integrate both sides of the equation separately. Integrating the left side, we get \( \int \frac{1}{y} \, dy = \ln |y| + C_1 \). For the right side, integrating \( \int 4x^3 \, dx \) gives \( x^4 + C_2 \).

Step 3: Simplify and Solve for y

Combine the constants into a single constant for simplicity. We get \( \ln |y| = x^4 + C \), where \( C = C_2 - C_1 \). To solve for \( y \), exponentiate both sides: \( |y| = e^{x^4 + C} = e^C e^{x^4} \). Let \( e^C = C' \), and since \( y \) can be negative or positive, the solution is \( y = C'e^{x^4} \).

Key Concepts

Separation of Variables

Separation of variables is a fundamental technique to solve differential equations. It involves rearranging the equation so that each variable and its differential are isolated on different sides of the equation. By doing this, you make it easier to integrate both sides separately.

• First, start with the original equation. In our example, we have \(\frac{dy}{dx} = 4x^3y\).
• Next, separate the variables to get terms involving \(y\) and \(dy\) on one side and terms involving \(x\) and \(dx\) on the other side. This gives us \(\frac{dy}{y} = 4x^3 dx\).

Separation doesn't solve the equation outright, but it makes the problem simpler by preparing it for the next step: integration. It's like setting up pieces of a puzzle so that they are easier to fit together.

Integration

Integration is a key calculus operation used to solve differential equations once variables are separated. If we have an equation in the form of \(\frac{dy}{y} = 4x^3 dx\), the next step is to integrate both sides independently.

• On the left, integrate \(\int \frac{1}{y} \, dy\), which results in \(\ln |y| + C_1\).
• On the right side, integrate \(\int 4x^3 \, dx\), leading to \(x^4 + C_2\).

Integration transforms the differential form into expressions we can manipulate algebraically. These indefinite integrals produce constants of integration \(C_1\) and \(C_2\), which can be merged into a single constant \(C\) for simplicity: \(\ln |y| = x^4 + C\). This step moves us closer to finding the general solution.

Exponential Functions

Exponential functions appear frequently when solving differential equations, especially after integrating logarithmic expressions. Continuing from our integrated form \(\ln |y| = x^4 + C\), we need to solve for \(y\) by eliminating the logarithm.

• Exponentiate both sides to get rid of the natural logarithm: \( |y| = e^{x^4 + C} \).
• This separates into \( |y| = e^C e^{x^4} \). By simplifying, let \( e^C = C' \) to form \( y = C' e^{x^4} \), where \(C'\) is any constant.

Exponential functions model growth and decay processes commonly observed in physics, biology, and economics, making this knowledge crucial for practical applications. Our final expression \(y = C' e^{x^4}\) is a general solution showing how \(y\) depends on \(x\).

Calculus

Calculus is the branch of mathematics that deals with change and motion, providing tools for analyzing functions and their rates of change. When working with differential equations, calculus helps us,

• understand the relationship between variables,
• isolate and integrate terms using techniques like separation of variables, and
• move from differential expressions to concrete solutions that describe real-world phenomena.

In our example, calculus allows us to take the differential equation \(\frac{dy}{dx}=4x^3y\) and process it step-by-step until we arrive at the general solution \( y = C' e^{x^4} \). This structured approach lets us navigate complex problems by breaking them into manageable pieces, providing insights into how functions behave and influence one another over their domains.