Problem 74 - Thomas Calculus

Question

Solve the initial value problems in Exercises \(71-90\) . $$\frac{d y}{d x}=9 x^{2}-4 x+5, \quad y(-1)=0$$

Step-by-Step Solution

Step 1: Identify the Differential Equation

The given differential equation is \( \frac{d y}{d x}=9x^{2}-4x+5 \). This is a first-order ordinary differential equation that can be solved using integration.

Step 2: Integrate the Differential Equation

To find \( y \), we need to integrate the right-hand side of the differential equation. \[ \int (9x^2 - 4x + 5) \, dx = 9 \int x^2 \, dx - 4 \int x \, dx + 5 \int 1 \, dx \]

Key Concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are a fundamental part of calculus and describe relationships involving functions of one independent variable and their derivatives. In simpler terms, an ODE involves equations where the rate of change of a variable relates to the variable itself. These equations are 'ordinary' because they deal with functions of a single variable, often either time or space. ODEs can model a vast range of physical phenomena like motion, growth, and decay processes.

• They typically appear in the form of \( \frac{dy}{dx} = f(x, y) \), which relates \( y \) (a function) and its derivative.
• Identifying an ODE is the first step in solving it, where our goal is finding a relation between \( y \) and \( x \).

Understanding ODEs is critical, as they are commonly applied in physics, engineering, and economics to solve real-world problems.

Integration

Integration is the process of finding a function given its derivative. It is essentially the reverse operation of differentiation and is used extensively to solve differential equations. In the context of solving ODEs, integration allows us to find the original function from its rate of change.

• The integral of a function \( f(x) \) with respect to \( x \) is written as \( \int f(x) \, dx \).
• In the solution step: \[ \int (9x^2 - 4x + 5) \, dx = 9 \int x^2 \, dx - 4 \int x \, dx + 5 \int 1 \, dx \]

Integration can be indefinite, where we seek a general function plus a constant \( C \), or definite, where we evaluate a function's change over an interval. Integrating correctly is key to finding the solution of ODEs.

First-Order Differential Equations

First-order differential equations are ODEs involving the first derivative of the function but not its higher derivatives. They take the general form \( \frac{dy}{dx} = f(x, y) \) and often describe dynamics where the rate of change at a point depends only on the current state.
These equations can be linear or non-linear:

• Linear first-order ODEs have the form \( \frac{dy}{dx} + P(x)y = Q(x) \).
• Non-linear does not fit this structure, like the exercise example \( \frac{dy}{dx} = 9x^2 - 4x + 5 \).

Solving these involves integrating the equation to replace the derivative with a function. Recognizing the type of first-order differential equation is vital as it determines the method used for solution.

Initial Condition

An initial condition helps uniquely determine the solution to a differential equation. It provides the exact value of the function at a particular point, which allows us to solve for any constants introduced by integration.
In our exercise example, the initial condition is \( y(-1) = 0 \).

• By using this, we adjust our integrated solution to pass through this point, giving us a specific solution rather than a family of solutions.
• After integrating, such conditions allow us to solve for unknown constants.

Initial conditions are crucial for practical problems as they ensure the mathematical model accurately represents a real-world situation.