Problem 21 - 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 . ~. $$ \frac{d y}{d x}=(x+1) y+(x+1) $$

Step-by-Step Solution

Answer

Use the integrating factor to solve the equation: \( y = F(x) \cdot e^{(\frac{x^2}{2} + x)} + C \cdot e^{(\frac{x^2}{2} + x)} \).

Step 1: Identify the type of differential equation

The given differential equation is \( \frac{dy}{dx} = (x+1)y + (x+1) \). This can be rewritten in standard linear form as \( \frac{dy}{dx} - (x+1)y = (x+1) \), which clearly indicates it's a first-order linear differential equation.

Step 2: Choose the method to solve

Since it's a linear first-order differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \), it can be solved using the integrating factor method. Here, \( P(x) = -(x+1) \) and \( Q(x) = (x+1) \).

Step 3: Calculate the integrating factor

The integrating factor \( \mu(x) \) is given by \( e^{\int P(x) \, dx} = e^{-\int (x+1) \, dx} = e^{-(\frac{x^2}{2} + x)} \).

Step 4: Multiply through by the integrating factor

Multiply every term in the differential equation by the integrating factor \( e^{-(\frac{x^2}{2} + x)} \). This gives: \( e^{-(\frac{x^2}{2} + x)} \frac{dy}{dx} - e^{-(\frac{x^2}{2} + x)} (x+1)y = e^{-(\frac{x^2}{2} + x)} (x+1) \).

Step 5: Simplify and integrate

Recognize that the left side of the equation is the derivative of \( y \cdot e^{-(\frac{x^2}{2} + x)} \). Thus, integrating both sides with respect to \( x \) gives \( y \cdot e^{-(\frac{x^2}{2} + x)} = \int e^{-(\frac{x^2}{2} + x)} (x+1) \, dx \).

Step 6: Integrate the right-hand side

To integrate \( \int e^{-(\frac{x^2}{2} + x)} (x+1) \, dx \), use integration techniques such as substitution or by parts as appropriate. Let's assume this integral evaluates to some function \( F(x) \).

Step 7: Solve for y

Multiply through by \( e^{(\frac{x^2}{2} + x)} \) to solve for \( y \), giving \( y = F(x) \cdot e^{(\frac{x^2}{2} + x)} + C \cdot e^{(\frac{x^2}{2} + x)} \), where \( C \) is the constant of integration.

Step 8: Verify the solution

Differentiate your solution for \( y \) and substitute back into the original equation to verify that it satisfies the given differential equation.

Key Concepts

Integrating Factor Method

The integrating factor method is a powerful technique to solve first-order linear differential equations. It is particularly useful when the equation cannot be easily separated into variables. The method involves the following essential steps:

• Identify a differential equation of the form \( \frac{dy}{dx} + P(x)y = Q(x) \).
• Calculate the integrating factor \( \mu(x) \), using the formula: \( \mu(x) = e^{\int P(x) \, dx} \).
• Multiply every term in the original equation by this integrating factor.
• Recognize the left-hand side as the derivative of \( y \cdot \mu(x) \) and integrate both sides to find a solution.

This method essentially transforms the original equation into a simpler form, making it easier to solve by integrating. The solution obtained can then be expressed in terms of a general function plus a constant of integration. This approach is crucial for handling equations that do not permit separation of variables.

First-Order Linear Differential Equation

A first-order linear differential equation is an equation involving an unknown function, its first derivative, and a first-degree polynomial form. A standard form for this equation is:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]

• \( P(x) \) and \( Q(x) \) are functions of \( x \) alone.

Such equations are common in modeling natural phenomena like growth and decay processes. Solving these equations generally involves finding a specific technique that exploits their linearity property, such as the integrating factor method. Recognizing the structure of these equations is the first step towards solving them effectively.

Separation of Variables

Separation of variables is another vital technique for solving differential equations. While it's applicable to a specific type, it simplifies problems immensely when applicable. This method works best when you can express the equation as a product of functions each dependent solely on one variable, typically in the form:
\[ g(y) \cdot dy = f(x) \cdot dx \]
The steps involve:

• Rearranging terms to isolate functions of \( y \) and \( x \) on opposite sides.
• Integrating both sides with respect to their respective variables.
• Combining the integrated functions to find a general solution.

Separation of variables is a straightforward, yet powerful technique when applicable, often making complex problems surprisingly simple to solve. However, it is not always suitable for equations, like the one we considered, where the integrating factor method was necessary.

Integration Techniques

Integration techniques are crucial when solving differential equations, especially when dealing with integrals that are not straightforward to compute. Some common methods include:

• Substitution: Used when an integral contains a composite function. It simplifies the integrand by substituting part of it with a single variable.
• Integration by parts: Applicable when the integrand is a product of two functions, allowing integration by using the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fraction Decomposition: A method to integrate rational functions, breaking them into simpler fractions that are easier to handle.

These techniques are employed based on the structure of the integral at hand. For instance, when solving the given differential equation, the integrating factor led us to an integral that required substitution or integration by parts to solve. Mastery of these techniques is essential for tackling complex differential equations effectively.