Question

Find the general solution to the differential equation.\(\frac{d x}{d t}=3 t^{2}\left(x^{2}+4\right)\)

Short answer

The general solution is \( x = 2 \tan(2t^3 + 2C) \).

Step-by-step solution

Recognize the Type of Differential Equation

The given differential equation \( \frac{dx}{dt} = 3t^2(x^2 + 4) \) is a separable differential equation. This means that we can rewrite it in a form that separates variables \( x \) and \( t \).

Separate Variables

Rewrite the equation to separate \( dx \) and \( dt \):\[ \frac{dx}{x^2 + 4} = 3t^2 \, dt \] Now, all \( x \) terms are on one side, and all \( t \) terms are on the other.

Integrate Both Sides

Integrate both sides of the equation:\[ \int \frac{dx}{x^2 + 4} = \int 3t^2 \, dt \] The left side integrates to \( \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C_1 \), and the right side integrates to \( t^3 + C_2 \).

Solve for the General Solution

Equating the two integrals results in:\[ \frac{1}{2} \arctan\left(\frac{x}{2}\right) = t^3 + C \]Where \( C = C_2 - C_1 \). Multiply both sides by 2:\[ \arctan\left(\frac{x}{2}\right) = 2t^3 + 2C \]

Isolation of \( x

Solve for \( x \) from the equation:\[ \frac{x}{2} = \tan(2t^3 + 2C) \]Thus, the general solution for \( x \) is:\[ x = 2 \tan(2t^3 + 2C) \]

Key concepts

General Solution

A general solution to a differential equation is an expression that contains a family of solutions. It usually includes an arbitrary constant because differential equations typically arise from problems with multiple possible solutions. In our exercise, the differential equation is separable, and the general solution involves expressing the dependent variable, here represented as \( x \), in terms of the independent variable, \( t \), along with one or more constants.
The form of the general solution, \( x = 2 \tan(2t^3 + 2C) \), reveals that there is a constant \( C \). This constant reflects the maximal generality of our solution. By varying \( C \), you can find different particular solutions, depending on initial conditions or other specific requirements.
When solving these equations, we first identify the type of differential equation. Here, it is separable, which means we can isolate \( x \) and \( t \) on different sides.
After successfully separating and integrating both sides, we solve the equation in terms of \( x \) to give a solution that covers any potential specific solution within the framework of the original differential equation.

Separation of Variables

Separation of variables is a key technique used to solve ordinary differential equations, especially those that can be written in a product form of two functions, one depending on \( x \) and the other on \( t \). This technique involves rearranging the equation

• Get all the \( x \)-related terms on one side
• Place \( t \)-related terms on the other side

With our specific differential equation given as \( \frac{dx}{dt} = 3t^2 (x^2 + 4) \), we rearrange it to \( \frac{dx}{x^2 + 4} = 3t^2 \, dt \).
After separation, each side of the equation only involves one variable. This positions us perfectly to apply integration, the natural next step in solving the separable differential equation.
The concept here is to tackle each variable independently, turning a complex differential equation into a solvable form using basic calculus on each side independently.
Separation simplifies the process, laying groundwork to make integration straightforward and effective.

Integration Techniques

Once the variables are separated, the next step is to integrate both sides of the equation. This is where integration techniques become crucial.
In our example, we have to integrate \( \int \frac{dx}{x^2 + 4} \) and \( \int 3t^2 \, dt \).

• The left side requires trigonometric substitution or an understanding of inverse trigonometric functions, yielding the result \( \frac{1}{2} \arctan\left(\frac{x}{2}\right) + C_1 \).
• The right side is simpler, leading to a basic polynomial integral which gives \( t^3 + C_2 \).

After integration, we equate both sides and solve for \( x \). The arbitrary constants \( C_1 \) and \( C_2 \) consolidate into a single constant \( C \) in our general solution.
Understanding these techniques allows us to move from a separated equation to a fully-developed general solution. Different integrals can require varying methods:

• Substitution
• Partial fractions
• Trigonometric identities

These tactics enrich the versatility of our approach, ensuring we have all the tools necessary to solve these equations effectively.