Problem 22 - Calculus for Biology and Medicine

Question

For each of the Problems 13-24 you should determine whether the problem needs to be solved using separation of variables or integrating factors (some of the problems may be solved using \mathrm{\\{} e i t h e r ~ m e t h o d ) . ~ T h e n ~ s o l v e ~ t h e ~ d i f f e r e n t i a l ~ e q u a t i o n . ~. $$ \frac{d y}{d x}=(x+1) y+(x+1) y^{2} $$

Step-by-Step Solution

Answer

Solve using separation of variables: \( y = \frac{Ce^{\frac{x^2}{2} + x}}{1 - Ce^{\frac{x^2}{2} + x}} \).

Step 1: Identify the Method

The given differential equation is \( \frac{dy}{dx} = (x+1)y + (x+1)y^2 \). It can be rewritten as \( \frac{dy}{dx} = (x+1)(y + y^2) \). Since both terms on the right-hand side share a common factor \((x+1)\), the equation can be rearranged to facilitate separation of variables. Therefore, the method to use is separation of variables.

Step 2: Separate Variables

Rewrite the equation in a form that separates the variables: \( \frac{dy}{y+y^2} = (x+1) dx \). To achieve this, factor \( y+y^2 \) as \( y(1+y) \), so the separated equation becomes: \( \frac{dy}{y(1+y)} = (x+1) dx \).

Step 3: Integrate Both Sides

Now, integrate both sides. Start with the left side: \( \int \frac{1}{y(1+y)} \, dy \). Perform partial fraction decomposition on \( \frac{1}{y(1+y)} \) as \( \frac{A}{y} + \frac{B}{1+y} \). Solving this gives \( \frac{1}{y(1+y)} = \frac{1}{y} - \frac{1}{1+y} \). Integrate to get \( \ln |y| - \ln |1+y| \). For the right side, integrate \( \int (x+1) \, dx = \frac{x^2}{2} + x + C \), where \( C \) is the constant of integration.

Step 4: Combine results and solve for y

The integrated form becomes \( \ln \left|\frac{y}{1+y}\right| = \frac{x^2}{2} + x + C \). Solve for \( y \) by exponentiating both sides: \( \frac{y}{1+y} = Ce^{\frac{x^2}{2} + x} \). Rearrange to solve for \( y \): \( y = \frac{Ce^{\frac{x^2}{2} + x}}{1 - Ce^{\frac{x^2}{2} + x}} \).

Key Concepts

Separation of Variables

Separation of variables is a mathematical method for solving ordinary differential equations. This technique involves rearranging an equation to isolate all the terms involving one variable on one side of the equation and all the terms involving the other variable on the opposite side.

For example, given the differential equation \( \frac{dy}{dx} = (x+1)(y + y^2) \), the goal is to rewrite this so that each side depends only on one of the variables. Simply put, you isolate the terms that contain \( y \) on one side and those that contain \( x \) on the other side.

Key steps include:

• Identify common factors or terms that can facilitate separation.
• Rearrange the terms into a multiplication that allows both variables to be expressed separately (i.e., conversion such that \( f(y) \, dy = g(x) \, dx \)).
• Once the variables are separated, you integrate both sides, resulting in two integrals often dependent on basic functions like logarithms or polynomials.

This standardized approach simplifies the integration step, allowing the equation to be solved more straightforwardly, especially for handling nonlinear ordinary differential equations.

Integrating Factors

An integrating factor is a powerful method used for solving non-separable linear differential equations, especially first-order linear differential equations of the form \( \frac{dy}{dx} + P(x)y = Q(x) \). An integrating factor is a function that turns a non-exact differential equation into an exact one.

The core idea involves multiplying the entire differential equation by a strategically chosen function \( \mu(x) \) which is determined by the formula \( \mu(x) = e^{\int P(x) \, dx} \).

Using integrating factors involves these steps:

• Identify \( P(x) \) from the standard form of the differential equation.
• Calculate the integrating factor \( \mu(x) \).
• Multiply every term in the equation by this integrating factor.
• Notice that the left-hand side of the equation becomes the derivative of a product \( \mu(x)y \), which simplifies the equation and makes it possible to integrate both sides easily.

This technique is particularly useful because it systematically transforms the differential equation into integrable form.

Partial Fraction Decomposition

Partial fraction decomposition is an algebraic tool used to break down complex rational expressions into simpler terms, especially useful in integration.

In the integration process of differential equations, you may encounter fractions that are difficult to integrate directly. Decomposition comes into play by expressing the fraction as a sum of simpler fractions whose denominators are linear or irreducible quadratic.

Steps include:

• Identify the fraction in your equation, like \( \frac{1}{y(1+y)} \).
• Assume a form of partial fractions, such as \( \frac{A}{y} + \frac{B}{1+y} \).
• Solve for the constants \( A \) and \( B \) by equating terms or substituting convenient values.
• Once determined, rewrite the fraction with these simpler components and integrate each term individually.

Partial fraction decomposition is essential for simplifying the integration process, allowing more complicated expressions to be managed with basic integration techniques.