Problem 15 - 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 . ~. $$ \frac{d y}{d t}=y^{2} t+y^{2} $$

Step-by-Step Solution

Answer

Use separation of variables, then integrate to find \( y = -\frac{1}{\frac{t^2}{2} + t + C} \).

Step 1: Identify the Type of Differential Equation

The given differential equation is \( \frac{dy}{dt} = y^2 t + y^2 \). The right-hand side can be factored as \( y^2(t + 1) \), so we see that both \( y \) and \( t \) terms are involved. It is a first-order ordinary differential equation (ODE).

Step 2: Choose the Method for Solving

Since the differential equation is expressed in terms of separable functions of \( y \) and \( t \), it is suitable for the method of separation of variables. This means we need to separate the variables \( y \) and \( t \) on opposite sides of the equation.

Step 3: Separate the Variables

Rearrange the terms to isolate \( y \) and its differential on one side and \( t \) on the other side: \[ \frac{1}{y^2} \, dy = (t + 1) \, dt \].

Step 4: Integrate Both Sides

Integrate both sides to solve for \( y \): \[ \int \frac{1}{y^2} \, dy = \int (t + 1) \, dt \].The left integral becomes \( -\frac{1}{y} + C_1 \) after integration, and the right integral becomes \( \frac{t^2}{2} + t + C_2 \).

Step 5: Combine Integration Constants

Combine the integration constants \( C_1 \) and \( C_2 \) into a single constant \( C \):\[ -\frac{1}{y} = \frac{t^2}{2} + t + C \].

Step 6: Solve for \( y \)

To express \( y \) explicitly, solve for \( y \):\[ y = -\frac{1}{\frac{t^2}{2} + t + C} \].

Key Concepts

Separation of Variables

The separation of variables is a technique used to solve first-order ordinary differential equations. This approach relies on the ability to separate the variables involved, making the equation easier to integrate. In the exercise presented, the differential equation is given by \( \frac{dy}{dt} = y^2 t + y^2 \),which can be rearranged by factoring to \( y^2(t + 1) \). The next step in separation of variables is to isolate each variable and their differentials on opposite sides. Through algebraic manipulation, you rewrite this as:

• \( \frac{1}{y^2} \, dy = (t + 1) \, dt \)

By setting the terms in this way, each side of the equation now involves only one type of variable, making it ready to integrate. Once integrated, the problem then becomes one of evaluating the known integrals:

• \( \int \frac{1}{y^2} \, dy \) and
• \( \int (t + 1) \, dt \).

Separation of variables works best when you can easily split the equation into these distinct variable parts, simplifying the integration process.

Integrating Factors

Integrating factors are a useful tool primarily when dealing with linear first-order differential equations that are not easily separable. Though not applied in this exercise, understanding integrating factors can still be insightful. They are generally used when given a linear first-order ODE of the form:\( \frac{dy}{dt} + P(t)y = Q(t) \).The integrating factor, denoted as \( \mu(t) \), is determined by:\( \mu(t) = e^{\int P(t) \, dt} \).The purpose of applying an integrating factor is to transform the original differential equation into one that can be rewritten as an exact differential:\( \frac{d}{dt} [\mu(t)y] = \mu(t)Q(t) \).By solving this form, we find:

• First, calculate the integrating factor \( \mu(t) \).
• Multiply the entire differential equation by this factor.
• Integrate both sides to obtain the solution.

While separation of variables was suitable here, integrating factors are a powerful alternative for equations that fit the linear form.

First-Order Ordinary Differential Equation

First-order ordinary differential equations (ODEs) involve functions and their first derivatives. These equations are fundamental in fields such as physics, engineering, and applied mathematics. In the given exercise, \( \frac{dy}{dt} = y^2 t + y^2 \), you deal with the first derivative of the function \( y \) with respect to \( t \), which qualifies it as a first-order ODE.Such equations appear frequently in modeling real-world phenomena, where a rate of change depends linearly or non-linearly on the function's current state. The two primary methods to solve first-order ODEs include separation of variables and integrating factors.To identify a first-order ODE:

• Look for the highest derivative being first-degree.
• Check for terms on one side of the equation being functions of the derivative, and on the other side, terms involving the independent variable or the function itself.

Understanding these core aspects helps in classifying and deciding the appropriate solution method for an ODE, making them a vital part of differential equations study.