Problem 59
Question
Solve the initial-value problem. $$ \frac{d y}{d x}=3 x^{2}, \text { for } x \geq 0 \text { with } y(0)=1 $$
Step-by-Step Solution
Verified Answer
The solution is \( y(x) = x^3 + 1 \).
1Step 1: Understand the Problem
We are given a differential equation \( \frac{d y}{d x} = 3x^2 \) with the initial condition \( y(0) = 1 \). Our task is to find the function \( y(x) \) that satisfies both the differential equation and the initial condition.
2Step 2: Integrate the Differential Equation
To solve for \( y(x) \), we need to integrate \( \frac{d y}{d x} = 3x^2 \). After integrating both sides with respect to \( x \), we have \( y(x) = \int 3x^2 \, dx = x^3 + C \).
3Step 3: Apply the Initial Condition
With the general solution \( y(x) = x^3 + C \), we use the initial condition \( y(0) = 1 \) to find \( C \). Substituting \( x = 0 \) into the equation gives \( 1 = 0^3 + C \), so \( C = 1 \).
4Step 4: Write the Particular Solution
Replacing \( C \) with 1 in the general solution, we get \( y(x) = x^3 + 1 \). This is the particular solution that satisfies the given initial-value problem.
Key Concepts
Understanding Differential EquationsThe Role of IntegrationUnderstanding Initial Conditions
Understanding Differential Equations
Differential equations are equations that involve the derivatives of a function. They describe how a particular quantity changes over time or space. In the given problem, we have the differential equation \( \frac{d y}{d x} = 3x^2 \). This equation tells us that the rate of change of \( y \) with respect to \( x \) is equal to \( 3x^2 \).
Differential equations are fundamental in modeling real-world situations such as motion, heat, and waves. They allow us to understand how a system evolves by defining the relationship between its variables and their rates of change. In many cases, the solution to a differential equation provides a function that shows the state of the system at any given point in time or space.
To solve a differential equation, one must find the function or set of functions that satisfy the given equation. This often involves techniques such as integration, especially for simple first-order differential equations like the one in our problem.
Differential equations are fundamental in modeling real-world situations such as motion, heat, and waves. They allow us to understand how a system evolves by defining the relationship between its variables and their rates of change. In many cases, the solution to a differential equation provides a function that shows the state of the system at any given point in time or space.
To solve a differential equation, one must find the function or set of functions that satisfy the given equation. This often involves techniques such as integration, especially for simple first-order differential equations like the one in our problem.
The Role of Integration
Integration is the inverse process of differentiation. When we have a differential equation and want to find a function that satisfies it, integration helps us reverse differentiate to find that original function. In our exercise, the differential equation is \( \frac{d y}{d x} = 3x^2 \). Integrating \( 3x^2 \) with respect to \( x \) provides us with the function \( y(x) \).
The process of integration involves finding the antiderivative. For \( 3x^2 \), the antiderivative is \( x^3 + C \), where \( C \) is a constant of integration. This arises because the derivative of a constant is zero, and thus integration can introduce an arbitrary constant.
In problems involving initial conditions, this constant can be determined by substituting the known values into the general solution. Solving these integrals helps us transition from the rate of change of a function to the actual function itself, crafting a picture of the dynamics described by the differential equation.
The process of integration involves finding the antiderivative. For \( 3x^2 \), the antiderivative is \( x^3 + C \), where \( C \) is a constant of integration. This arises because the derivative of a constant is zero, and thus integration can introduce an arbitrary constant.
In problems involving initial conditions, this constant can be determined by substituting the known values into the general solution. Solving these integrals helps us transition from the rate of change of a function to the actual function itself, crafting a picture of the dynamics described by the differential equation.
Understanding Initial Conditions
An initial condition is an extra piece of information provided alongside a differential equation to help find a specific solution. In differential equations, there can be many potential solutions, or a family of solutions, due to the constant of integration introduced during integration.
In our problem, the initial condition is given as \( y(0) = 1 \). Initial conditions are crucial because they help us determine a unique solution from the general solution.
By substituting the initial condition into our general solution \( y(x) = x^3 + C \), we substitute \( 0 \) for \( x \) and set the equation equal to \( 1 \). This helps us find that \( C = 1 \). Substituting back gives us the particular solution \( y(x) = x^3 + 1 \). This particular solution satisfies both the differential equation and the initial condition, specifying the exact behavior of the function at the starting point defined by the initial condition.
Initial conditions are vital in engineering, physics, and other sciences because they allow for the prediction and understanding of specific behaviors under given scenarios.
In our problem, the initial condition is given as \( y(0) = 1 \). Initial conditions are crucial because they help us determine a unique solution from the general solution.
By substituting the initial condition into our general solution \( y(x) = x^3 + C \), we substitute \( 0 \) for \( x \) and set the equation equal to \( 1 \). This helps us find that \( C = 1 \). Substituting back gives us the particular solution \( y(x) = x^3 + 1 \). This particular solution satisfies both the differential equation and the initial condition, specifying the exact behavior of the function at the starting point defined by the initial condition.
Initial conditions are vital in engineering, physics, and other sciences because they allow for the prediction and understanding of specific behaviors under given scenarios.
Other exercises in this chapter
Problem 58
Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{1}{x+1}, x>-1 $$
View solution Problem 58
Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow 0^{+}}\left(\frac{1}{x}-\frac{1}{\sqrt{x}}\right) $$
View solution Problem 59
Find the limits in Problems 1-60; not all limits require use of l'Hôpital's rule. $$ \lim _{x \rightarrow+\infty} \frac{\sqrt{x+1}}{\sqrt{x}} $$
View solution Problem 60
Solve the initial-value problem. $$ \frac{d y}{d x}=\frac{x^{2}}{3}, \text { for } x \geq 0 \text { with } y(0)=2 $$
View solution