Problem 2 - Applied Calculus

Question

Find particular solutions \(\frac{d y}{d t}=0.5(y-200), \quad y=50\) when \(t=0\)

Step-by-Step Solution

Answer

The particular solution is \( y(t) = 200 - 150e^{0.5t} \).

Step 1: Identify type of differential equation

The given differential equation is \( \frac{dy}{dt} = 0.5(y - 200) \). This is a first-order linear ordinary differential equation of the form \( \frac{dy}{dt} = ky + c \), where \( k = 0.5 \) and \( c = -100 \).

Step 2: Solve the homogeneous equation

The homogeneous part of the equation is \( \frac{dy}{dt} = 0.5y \). We solve this by separation of variables: \( \int \frac{1}{y} \, dy = \int 0.5 \, dt \), leading to \( \ln |y| = 0.5t + C_1 \). By exponentiating both sides, we get \( y = C e^{0.5t} \) as the general solution to the homogeneous equation.

Step 3: Solve for particular solution using initial conditions

Substitute the initial condition \( y(0) = 50 \) into the general solution \( y(t) = C e^{0.5t} + 200 \). At \( t = 0 \), the equation becomes \( 50 = C e^{0} + 200 \), which simplifies to \( 50 = C + 200 \). Solve for \( C \) to get \( C = -150 \).

Step 4: Write the particular solution

Substituting \( C = -150 \) back into the general solution form, the particular solution is \( y(t) = 200 - 150e^{0.5t} \).

Key Concepts

First-order Differential Equations

First-order differential equations are a type of differential equation where the highest derivative is the first derivative. In these equations, we are mainly concerned with how a function changes with respect to another variable. In our exercise, this is represented by \(\frac{dy}{dt} = 0.5(y-200)\).

A first-order differential equation often has the form \(a(t)y' + b(t)y = f(t)\), where \(y'\) is the first derivative of \(y\) with respect to \(t\). For linear equations, like the one we are solving, the function \(f(t)\) might consist solely of the dependent variable \(y\) and constants.

To solve these equations, we utilize different methods such as separation of variables or integrating factors. Recognizing the type and structure of the equation allows us to apply the correct solution method. In our example, the problem is simplified because we can separate the variables to find the general solution.

Ordinary Differential Equations

Ordinary differential equations (ODEs) involve functions of a single variable and their derivatives. Unlike partial differential equations, which have multiple variables, ODEs are usually easier to solve and analyze. The exercise we're working on is an example of an ODE. It describes a situation where the rate of change of a function is directly related to the function itself and a constant.

Ordinary differential equations are categorized by "order," based on the highest derivative present in the equation. In our problem, it's a first-order ODE because it involves \(\frac{dy}{dt}\), the first derivative of \(y\) with respect to \(t\). Additionally, this ODE is linear since it conforms to the standard linear form.

Understanding ODEs is essential because they model a wide range of real-world phenomena, such as population growth, radioactive decay, and in this case, the cooling or heating process over time.

Initial Value Problem

An initial value problem is a type of differential equation problem that, in addition to the differential equation itself, provides a specific value of the function at a certain point, called the initial condition. This is crucial because it allows us to find a particular solution that fits the initial conditions given.

In the given exercise, we have the initial condition \(y(0) = 50\), which means that at time \(t = 0\), the value of \(y\) is 50. This initial condition helps specifically determine the constant \(C\) in the equation \(y = Ce^{0.5t} + 200\). Solving for \(C\) using the initial condition gives us the particular solution \(y(t) = 200 - 150e^{0.5t}\).

Initial value problems are fundamental in verifying real-world applications, as they provide the means to tailor general solutions to fit specific scenarios, thereby enabling accurate predictions and insights about the behavior of dynamic systems over time.