Problem 17 - 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 . ~. $$ \frac{d y}{d t}=\cos t $$

Step-by-Step Solution

Answer

The general solution is \( y = \sin t + C \).

Step 1: Identify the Type of Differential Equation

The given equation is \( \frac{dy}{dt} = \cos t \). This is a first-order ordinary differential equation. We need to determine the method to solve it: separation of variables or integrating factors.

Step 2: Choose the Appropriate Method

The right-hand side of the equation \( \cos t \) is purely a function of \( t \) and does not involve \( y \). This suggests that separation of variables is not needed. Instead, the differential equation can be solved directly by integration since it is already in the form \( \frac{dy}{dt} = f(t) \).

Step 3: Integrate Both Sides

To solve \( \frac{dy}{dt} = \cos t \), integrate both sides with respect to \( t \):\[ y = \int \cos t \, dt \]

Step 4: Compute the Integral

The integral of \( \cos t \) with respect to \( t \) is \( \sin t \). Therefore, we have:\[ y = \sin t + C \]where \( C \) is the constant of integration.

Step 5: Present the General Solution

Since there are no initial conditions given, the general solution to the differential equation is:\[ y = \sin t + C \]

Key Concepts

Separation of Variables

Separation of variables is a powerful method used to solve certain types of ordinary differential equations, specifically those where variables can be moved to separate sides of the equation. The key idea is to rearrange the differential equation such that all terms involving one variable are on one side, and all terms involving the other variable are on the other side. Here’s how it works:

• Start with an equation that can be expressed in the form \( \frac{dy}{dx} = g(y)h(x) \).
• Rearrange it to get \( \frac{dy}{g(y)} = h(x)dx \).
• Integrate both sides separately, with respect to their own variables.

This technique is effective when the equation aligns with the structure allowing the separation of the variables. However, it is not applicable in cases where the dependent variable cannot be isolated to one side or when the right-hand side is a function only of time, as in our exercise with the differential equation \( \frac{dy}{dt} = \cos t \). In this case, the separation of variables is not employed, because the integrable form is already set directly based on time.

Integrating Factors

Integrating factors are used to solve linear first-order differential equations that might not be easily separable. This method involves multiplying the entire differential equation by a carefully chosen function, called the integrating factor, to make the equation integrable.

Here’s the step-by-step approach for using integrating factors:

• Consider a linear equation in the form \( \frac{dy}{dt} + P(t)y = Q(t) \).
• Identify the integrating factor \( \mu(t) = e^{\int P(t) dt} \).
• Multiply the entire equation by \( \mu(t) \) making it into a form where the left-hand side is the derivative \( \frac{d}{dt}[\mu(t) y] \).
• Integrate both sides with respect to time \( t \) to solve for \( y(t) \).

In our exercise, since the differential equation \( \frac{dy}{dt} = \cos t \) is already in the form that directly integrates \( f(t) \), the use of an integrating factor is not necessary. It is important to recognize when this method could streamline integration, especially for equations that are not as straightforward.

Ordinary Differential Equations

Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives. They are called "ordinary" to differentiate them from partial differential equations, which involve multiple variables. ODEs are fundamentally important in math, engineering, and the sciences because they model tons of real-world systems that evolve over time, like population dynamics or electrical circuits.

The equation given in the exercise, \( \frac{dy}{dt} = \cos t \), is an example of a first-order ODE because it involves the first derivative of \( y \) with respect to \( t \).

• First-order ODEs take the general form \( \frac{dy}{dt} = f(t, y) \), where \( f \) can be dependent on both \( t \) and \( y \).
• They can often be solved through integration if \( f(t, y) \) is simple, or using advanced methods like separation of variables or integrating factors if more complex.

Understanding how to classify and solve these types of equations is crucial for analyzing systems modeled by ODEs. These tools give you the ability to solve for functions that describe change over time, which is a powerful skill in scientific and engineering disciplines.