Problem 22 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Solve the differential equations below. Find the general solution. (a) \(\frac{d y}{d t}=\sin 3 t\) (b) \(\frac{d y}{d t}=5 \cdot 2^{t}\) (c) \(\frac{d x}{d t}=\frac{t+1}{l}\) (d) \(\frac{d x}{d t}=\frac{t+1}{t^{2}}\)

Step-by-Step Solution

Answer

The general solutions are (a) \(y(t) = -\frac{1}{3}\cos(3t) + C\), (b) \(y(t) = \frac{5 \cdot 2^{t}}{\ln(2)} + C\), (c) \(x(t) = \frac{t^2}{2l} +\frac{t}{l} + C\), and (d) \(x(t) = -\frac{1}{t} + \ln |t| + C\).

Step 1: Solve equation (a)

The equation \(\frac{d y}{d t}=\sin 3 t\) can be integrated directly. The integral of the right-hand side will give the function \(y(t)\). This yields \(y(t) = -\frac{1}{3}\cos(3t) + C\), where \(C\) is the constant of integration.

Step 2: Solve equation (b)

For the differential equation \(\frac{d y}{d t}=5 \cdot 2^{t}\), you use the rule of exponential integration. The integral of \(b^x\) with respect to \(x\) is \(\frac{b^x}{\ln(b)}\). Thus, \(y(t) = \frac{5\cdot 2^{t}}{\ln(2)} + C\).

Step 3: Solve equation (c)

The differential equation \(\frac{d x}{d t}=\frac{t+1}{l}\) can be integrated directly. By doing so, you get \(x(t) = \frac{t^2}{2l} +\frac{t}{l} + C\).

Step 4: Solve equation (d)

The last differential equation \(\frac{d x}{d t}=\frac{t+1}{t^{2}}\) is a bit more challenging having to use partial fraction decomposition method first. Simplifying, \(\frac{1}{t^2} + \frac{1}{t}\), and then integrating separately gives \(x(t) = -\frac{1}{t} + \ln |t| + C\).

Key Concepts

Integral Calculus

Integral calculus is all about finding the integral of functions, which gives us the area under the curve of a graph of a function. It's the opposite of differentiation. Consider the formula for integration: if you have a function \( f(t) \) and you find its integral, you're essentially reversing the process of finding the derivative.

• To integrate, constants like \( C \) come into play, known as the constant of integration. They account for any vertical shifts in the function.
• For example, integrating \( \sin(3t) \) results in \( y(t) = -\frac{1}{3}\cos(3t) + C \).

Understanding integral calculus is crucial for solving differential equations and allows us to explore quantities like areas, volumes, and other quantities that accumulate over time.

Exponential Functions

Exponential functions are characterized by their constant rate of growth; they appear in various fields such as biology, finance, and physics. The general form of an exponential function is \( a \cdot b^x \), where \( a \) is a constant and \( b \) is the base of the exponential.

• In the context of differential equations, exponential functions arise naturally when modeling growth processes. For example, \( \frac{d y}{d t}=5 \cdot 2^{t} \) is an exponential function.
• The integral of an exponential function \( b^x \) with respect to \( x \) is \( \frac{b^x}{\ln(b)} + C \).

Recognizing how exponential functions behave and how to integrate them is essential for solving various real-world problems.

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational expressions into simpler fractions, making them easier to integrate. This method is especially useful when dealing with fractions that have quadratic or higher degree polynomials in the denominator.

• Take, for instance, the expression \( \frac{t+1}{t^2} \). By breaking it down, it simplifies to \( \frac{1}{t^2} + \frac{1}{t} \).
• This allows us to integrate each term separately, simplifying the integration process into manageable steps.

Mastering partial fraction decomposition is valuable for tackling complex integrals and obtaining solutions efficiently.

Constant of Integration

When integrating, you often add a constant known as the constant of integration, denoted by \( C \). This constant reflects the fact that an indefinite integral represents a family of functions, not just a single solution.

• The constant \( C \) accounts for vertical shifts in the graph of the integrated function, ensuring all possible solutions are considered.
• For example, when integrating \( \sin(3t) \), the output is \( -\frac{1}{3}\cos(3t) + C \).

Understanding the role of the constant of integration is crucial for comprehending the general solution to a differential equation.