Question

Find the general solution to the differential equation.$\frac{dx}{dt}=3t^2(x^2+4)$

Short answer

The general solution is $\arctan \left(\frac{x}{2}\right)=2t^3+C$.

Step-by-step solution

Separate Variables

To solve the differential equation $\frac{dx}{dt}=3t^2(x^2+4)$, we begin by separating the variables. This means we want to have all terms involving $x$ on one side of the equation and all terms involving $t$ on the other. Divide both sides by $x^2+4$ and multiply both sides by $dt$:$$\frac{1}{x^2+4}dx=3t^{2}dt$$

Integrate Both Sides

The next step is to integrate both sides of the equation to find the general solution. Integrate the left side with respect to $x$ and the right side with respect to $t$:$$\int \frac{1}{x^2+4}dx=\int 3t^{2}dt$$

Integrate the Left Side

The integral on the left involves $\frac{1}{x^2+4}$. Recognize that the antiderivative of this expression is related to the arctangent function:$$\int \frac{1}{x^2+4}dx=\frac{1}{2}\arctan \left(\frac{x}{2}\right)+C_1$$

Integrate the Right Side

Now, integrate the right side of the equation, $\int 3t^{2}dt$:$$\int 3t^{2}dt=t^3+C_2$$

Combine and Simplify Results

Combine the results from both integrals to form the general equation:$$\frac{1}{2}\arctan \left(\frac{x}{2}\right)=t^3+C$$where $C=C_2-C_1$. Multiply both sides by 2 to simplify:$$\arctan \left(\frac{x}{2}\right)=2t^3+2C$$

Key concepts

Separation of Variables

Separation of variables is a powerful method used to solve differential equations. It involves rearranging the equation so that each variable appears on a separate side of the equation. When we have an equation such as $\frac{dx}{dt}=3t^2(x^2+4)$, our goal is to isolate $x$ on one side and $t$ on the other.
This is achieved by dividing both sides by $x^2+4$ and multiplying by $dt$.
Thus, we transform the original equation into:

• $\frac{1}{x^2+4}dx=3t^{2}dt$

This separates the variables and allows us to integrate each side separately.

This approach is useful for first-order differential equations, especially when:

• Each side can be integrated easily
• Solving the equation requires combining simple algebraic manipulations.

Mastering this technique opens the door to solving more complex systems with similar properties.

Integration Techniques

Integration techniques are key in differential equations to find solutions. After separating variables, the next step is to integrate both sides of the equation. Consider integrating $\frac{1}{x^2+4}dx$ and $3t^{2}dt$. Each requires a specific technique.

• For the left side, recognizing a standard form helps. The integral $\int \frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan \left(\frac{x}{a}\right)+C$ is used due to its link to the arctangent function.
• For the right side, the power rule for integrals \, $\int t^{n}dt=\frac{t^{n+1}}{n+1}+C$ \, works well for $3t^{2}dt$.

Different integration methods, like substitution or recognizing patterns, aid in solving various parts of an equation.
Once these integrals are evaluated, they are combined to form a general solution. Understanding these techniques is essential for the seamless transition from differential equations to finding their solutions.

Arctangent Function

The arctangent function, $\arctan (x)$, is deeply connected to integrals involving quadratic expressions. When dealing with $\frac{1}{x^2+a^2}$, the result of the integral involves the arctangent's inverse relationship, leading to solutions like $\frac{1}{a}\arctan \left(\frac{x}{a}\right)+C$.

Understanding the role of $\arctan (x)$ involves recognizing:

• The function's range is from $-\frac{\pi }{2}$ to $\frac{\pi }{2}$, reflecting how it maps real numbers to a finite range.
• It's useful for integrals where direct algebraic methods fail to simplify the result.
• Arctangent simplifies integrating functions with structure $\frac{1}{x^2+4}$, as seen in this exercise.

Thus, it bridges a gap between algebraic expressions and trigonometric functions during integration. By grasping its usage, you can tackle more complex integrals involving quadratic denominators with ease.