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Chapter 6: Problem 3

Identify the differential equation as one that can be solved using only antiderivatives or as one for which separation of variables is required. Then find a general solution for the differential equation. $$ \frac{d y}{d x}=6 x^{2} y $$

Short Answer

Expert verified
The differential equation can be solved using separation of variables. The general solution is \( y = Ce^{2x^3} \), where \( C \) is any constant.

Step by step solution

01

Identify the Type of Differential Equation

The given differential equation is \( \frac{dy}{dx} = 6x^2y \). This equation has the form \( \frac{dy}{dx} = f(x)y \), which suggests that it is a separable differential equation.
02

Set Up the Separation of Variables

To solve the equation by separation of variables, rewrite it as \( \frac{dy}{y} = 6x^2 \, dx \). This separates the \( y \) terms and \( x \) terms on different sides of the equation.
03

Integrate Both Sides

Integrate both sides of the separated equation. The left side integrates to \( \ln |y| \) and the right side integrates to \( 2x^3 + C \), leading to the equation \( \ln |y| = 2x^3 + C \).
04

Solve for y

To find an expression for \( y \), exponentiate both sides to get \( |y| = e^{2x^3 + C} \). Simplifying, we can write \( y = Ce^{2x^3} \) where \( C \) is an arbitrary constant, reflecting the general solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Separable Differential Equations
Separable differential equations are a type of differential equation where the variables can be separated on different sides of the equation. This means we can manipulate the equation so that all terms involving one variable, say \(y\), are on one side of the equation, and all terms involving the other variable, \(x\), are on the other side. An equation is separable if it can be written in the form:
  • \(\frac{dy}{dx} = g(x)h(y)\)
To solve it, we rearrange it to:
  • \(\frac{1}{h(y)} dy = g(x) dx\)
This allows us to integrate both sides independently with respect to their respective variables, which leads us to a solution.
Separable differential equations are often straightforward to solve, because the process reduces the integration into manageable parts, making it easier to find an analytical solution.
General Solution
The general solution of a differential equation is a solution that contains all possible solutions of the differential equation. This typically involves an arbitrary constant, often represented by \(C\). In our case, after we solved the differential equation \(\frac{dy}{dx} = 6x^2y\), we found the general solution to be:
  • \(y = Ce^{2x^3}\)
Here, \(C\) is an arbitrary constant that can take any value. This constant is crucial as it accounts for the infinite number of solutions that differ by a constant multiple.
General solutions are important because they provide the broadest form of the solution and can be refined into specific solutions by using initial conditions or additional data.
Integration in Calculus
Integration is a fundamental concept in calculus used for finding antiderivatives, also known as indefinite integrals. When solving separable differential equations, integration helps us reverse the process of differentiation to find functions whose derivatives meet the form of our equations. In this context:
  • The integral of \(\frac{1}{y} dy\) was \(\ln |y|\)
  • The integral of \(6x^2 dx\) resulted in \(2x^3\)
Integrating in calculus provides us with general solutions containing constant terms (like the \(C\) in our solution \(y = Ce^{2x^3}\)) that represent the set of all possible solutions.
Being able to perform integration skillfully is essential for solving many types of differential equations and applying solutions to real-world problems.

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