Problem 49 - Calculus for Biology and Medicine

Question

Find the general solution of the differential equation. $$ \frac{d y}{d x}=x(1+x), x>0 $$

Step-by-Step Solution

Answer

The general solution is \( y = \frac{x^2}{2} + \frac{x^3}{3} + C \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \frac{dy}{dx} = x(1+x) \). This is a first-order ordinary differential equation and it can be solved using the method of separation of variables.

Step 2: Separate Variables

Rewrite the equation so that all the terms involving \( y \) are on one side and terms involving \( x \) are on the other side:\[\frac{dy}{dx} = x + x^2\]Currently, there are no \( y \) terms, so the equation is ready for integration as is.

Step 3: Integrate Both Sides

Integrate both sides of the equation with respect to \( x \):\[\int dy = \int (x + x^2) \, dx\]This yields:\[y = \int x \, dx + \int x^2 \, dx\]

Step 4: Compute the Integrals

Calculate the integrals on the right side:\[\int x \, dx = \frac{x^2}{2} + C_1\]\[\int x^2 \, dx = \frac{x^3}{3} + C_2\]Combine these to get:\[y = \frac{x^2}{2} + \frac{x^3}{3} + C\]where \( C = C_1 + C_2 \) is a constant of integration.

Step 5: Write the General Solution

The solution to the differential equation is:\[y = \frac{x^2}{2} + \frac{x^3}{3} + C\]This is the general solution involving the arbitrary constant \( C \).

Key Concepts

Ordinary Differential Equations

An ordinary differential equation (ODE) is a mathematical equation involving functions of only one independent variable and its derivatives. For example, in the equation \( \frac{dy}{dx} = x(1+x) \), \( y \) is dependent on \( x \), and \( \frac{dy}{dx} \) represents the derivative of \( y \) with respect to \( x \). This equation is classified as a first-order ODE since it involves only the first derivative of \( y \). ODEs are crucial in modeling natural phenomena such as motion, growth, and decay in various fields like physics, biology, and engineering. Key features of ODEs include:

• An expression involving derivatives of unknown functions.
• A single independent variable.
• Solutions that provide a function or a set of functions.

By solving ODEs, we aim to find a general solution or a particular solution that satisfies initial or boundary conditions. In this context, the given ODE \( \frac{dy}{dx} = x(1+x) \) is solved to obtain a function \( y(x) \) that describes how \( y \) changes with \( x \).

Separation of Variables

Separation of variables is a powerful technique used to solve ordinary differential equations. It involves rearranging the equation to isolate variables on opposite sides of the equation. This prepares the equation for integration, a crucial step in finding the solution. In the equation \( \frac{dy}{dx} = x + x^2 \), the technique becomes straightforward because there are no \( y \) terms, allowing us to directly integrate with respect to \( x \). Typically, if the ODE involves both \( x \) and \( y \), you would rearrange it to the form \( g(y) \, dy = f(x) \, dx \), where each side can then be integrated separately. The primary steps are:

• Rearrange the equation to separate the variables.
• Integrate both sides individually.
• Find the integration constants to establish the general solution.

This method is particularly effective when the function can be expressed in separable terms, like the example here, enabling easy computation of the integrals. It's a strategy that simplifies and accelerates solving certain ODEs.

Integration

Integration is a mathematical process that determines the accumulated value or area under the curve of a function. It essentially reverses the process of differentiation. In solving the differential equation \( \frac{dy}{dx} = x + x^2 \), integration allows us to express \( y \) in terms of \( x \). By integrating each part of the right side of the equation separately, we find:

• The integral of \( x \) gives \( \frac{x^2}{2} \).
• The integral of \( x^2 \) provides \( \frac{x^3}{3} \).

Combining these results includes adding a constant \( C \), which arises since integration can yield an infinite set of potential solutions, shifted vertically on a graph.Here's a step-by-step look at the integration process in this context:

• Set up the integrals for each term: \( \int x \, dx \) and \( \int x^2 \, dx \).
• Using the power rule for integration, compute integrals: The power rule states \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
• Combine the results to write the general solution: \( y = \frac{x^2}{2} + \frac{x^3}{3} + C \).

Mastering integration is beneficial not only for solving ODEs but also for a broader range of applications in science and engineering, understanding it fundamentally enhances problem-solving skills.