Question

Identify each of the differential equations as type (for example, separable, linear first order, linear second order, etc.), and then solve it. $$d y-\left(2 y+y^{2} e^{3 x}\right) d x=0$$

Short answer

First-order nonlinear, solved by separating variables and integrating.

Step-by-step solution

Identify the type of differential equation

Rewrite the given differential equation in the form of \( \frac{dy}{dx} \). The equation is: \[ d y = \(2 y + y^2 e^{3x} \) dx \] \, which can be rewritten as \[ \frac{dy}{dx} = 2 y + y^2 e^{3x} \]. This is a first-order nonlinear differential equation since it cannot be written in the form of a linear first-order equation.

Check if the equation is separable

To check if the equation is separable, we see if it can be written in the form \[ \frac{dy}{dx} = g(y)h(x) \]. \ Rewrite the equation: \[ \frac{dy}{dx} = y \( 2 + y e^{3x} \) \]. \ Yes, this can be written as a separable equation since we can separate the variables y and x on each side: \[ \frac{1}{y \ (2 + y e^{3x})} dy = dx \].

Separate the variables

Separate the variables y and x: \[ \frac{1}{y \ (2 + y e^{3x})} dy = dx \].

Integrate both sides

Integrate both sides of the separated equation: \[ \int \frac{1}{y \ (2 + y e^{3x})} dy = \int dx \]. \ The left side of the equation may require partial fraction decomposition or another method for integration, which requires recognizing the form or appropriate substitution.

Use substitution method

Let \( v = 2 + y e^{3x} \), then \ d(v - 2) = y e^{3x} dx. Solve for y and express the equation in terms of v.

Integrate and simplify

Integrate and find integrating factor or appropriate antiderivative for both sides.

Obtain the general solution

Once both sides have been integrated and simplified, equalize them to obtain the general form of the solution.

Key concepts

separable differential equations

A separable differential equation can be written in the form \( \frac{dy}{dx} = g(y) h(x) \). This form allows us to separate the variables y and x on each side of the equation. Let’s break this down with an example.

Consider the differential equation provided in the exercise:
\[ \frac{dy}{dx} = y (2 + y e^{3x}) \]
To separate the variables, you can rewrite it as:
\[ \frac{1}{y (2 + y e^{3x})} dy = dx \]
By isolating y on one side and x on the other, we’ve successfully separated the variables. This allows us to integrate each side independently, which is a significant advantage in solving the equation.

nonlinear differential equations

Nonlinear differential equations are characterized by the fact that the dependent variable or its derivatives appear in a non-linear form.
For instance, in our given problem,
\[ \frac{dy}{dx} = 2 y + y^2 e^{3x} \]
The presence of the term \( y^2 \) indicates that this is a nonlinear differential equation. Unlike linear equations, these can be more complex to solve and often require special techniques or transformations.
Nonlinear equations might also have multiple solutions or exhibit behaviors such as chaos in certain contexts.
Identifying a nonlinear differential equation is crucial because it helps in determining the appropriate method to approach its solution.

integration techniques

In solving differential equations, integration plays a vital role. We often need various integration techniques to simplify and solve these equations. Here are some commonly used methods:

• Direct Integration: When the integrand is straightforward.
• Substitution Method: Useful when an integral can be simplified by substituting variables.
• Partial Fraction Decomposition: Breaks down a complex fraction into simpler ones.

In our exercise, we might start with the substitution method after separating the variables. Then, depending on the form of the integrand, we could use partial fractions or another suitable technique to find the solution.

substitution method

The substitution method is a powerful technique where we replace a variable or expression with another variable. This can simplify the equation and make it easier to integrate.
In our example:
\[ \int \frac{1}{y (2 + y e^{3x})} dy = \int dx \]
We let \( v = 2 + y e^{3x} \). Then, the equation can be rearranged in terms of v and its differential.
This often transforms a complex problem into a simpler one that we can solve using basic integration or other methods.

partial fraction decomposition

Partial fraction decomposition is a technique used for breaking complex rational fractions into simpler ones, making integration more manageable.
For example:
If we have a fraction like \[ \frac{1}{y (2 + y e^{3x})} \],
it can potentially be decomposed into simpler parts that are easier to integrate separately.
The fundamental idea is to express the complex fraction as a sum of simpler fractions with known integrals. This method is especially useful in handling equations resulting from separable differential equations.