Question

Solve each differential equation.$$\frac{d y}{d x}=2 x\left(x^{2}+6\right)$$

Short answer

\( y = \frac{1}{2}x^4 + 6x^2 + C \)

Step-by-step solution

Identify the Type of Differential Equation

Examine the given differential equation \( \frac{dy}{dx} = 2x(x^2 + 6) \) to identify its type. Here, the equation is a first-order ordinary differential equation that can be directly integrated since it is already solved for \(\frac{dy}{dx}\) and involves only a function of \(x\) on the right side.

Integrate Both Sides of the Equation

Integrate the equation with respect to \(x\) to find \(y\): \[ \int \frac{dy}{dx} dx = \int 2x(x^2 + 6) dx \] This will find the general solution for \(y\) in terms of \(x\).

Perform the Integration

Carry out the integration on the right side: \[ y = \int 2x^3 + 12x dx \] Distribute the constant and integrate each term separately: \[ y = \frac{1}{2} \int 2(2x^3) dx + 12 \int x dx = \frac{1}{2}(2x^4/2) + 12(x^2/2) + C \] Simplify: \[ y = \frac{1}{2}(x^4) + 6(x^2) + C \] where \(C\) is the constant of integration.

Key concepts

Ordinary Differential Equation

When we talk about an ordinary differential equation (ODE), we refer to an equation involving derivatives of a function with respect to one independent variable. ODEs are classified by their order, which is determined by the highest derivative present in the equation. The differential equation


The equation presented in the exercise, is a first-order ODE since the highest derivative it contains is . In this case, the ODE is also separable, making the solution straightforward through basic integration techniques.

Integration of Functions

The essence of solving an ODE often involves the integration of functions. Integration is the inverse process of differentiation, and it allows us to find a function given its derivative. In the context of differential equations, integration enables us to recover the original function from the rate of change. In the provided problem, we performed integration to reverse the differentiation that produced . You can think of integration as 'summing up' tiny pieces to find the whole.
Integration can be done using a variety of techniques, including power rule, substitution, and integration by parts. Our example used the power rule to integrate terms like and .
It's crucial to understand these techniques to handle a range of ODEs effectively.

Constant of Integration

When we integrate a function, we add a constant of integration (C). This is because indefinite integration, unlike differentiation, is not a unique operation. When you differentiate a constant, it becomes zero, which means different functions can have identical derivatives, differing only by a constant.

For example, if you take and , both differentiate to . So, when reversing the process through integration, we must include the constant of integration to account for all possible original functions. In our step by step solution, it was the last term we added after integrating.
This constant plays a crucial role, especially when solving initial value problems where it can be determined by given conditions to find a specific solution.