91Ó°ÊÓ

Implicit solutions are a way of expressing the solution to a differential equation without necessarily solving for one variable in terms of another. In this case, the implicit solution is given as an equation that relates the dependent variable and the independent variable. This can be very useful when the relationship between the variables is complex or the equation is difficult to rearrange.

In the given initial-value problem, after integrating both sides of the separated variables, we arrived at the equation:

• \( \tan^{-1}(x) = 4t + C \)

By applying the initial condition \( x(\pi/4) = 1 \), we calculated the constant \( C \) and found the implicit solution:

• \( \tan^{-1}(x) = 4t - \frac{3\pi}{4} \)

This implicit form of the solution allows us to see the relationship between \( x \) and \( t \) without isolating \( x \) on one side of the equation. Some equations are best left in this form due to their complexity or the nature of the relationship between \( x \) and \( t \).

In contrast to implicit solutions, explicit solutions provide a way to express the dependent variable directly in terms of the independent variable. For the initial-value problem in question, the explicit solution requires solving for \( x \) in terms of \( t \).

To find the explicit solution, we started with the implicit solution:

• \( \tan^{-1}(x) = 4t - \frac{3\pi}{4} \)

By taking the tangent of both sides, we isolate \( x \):

• \( x = \tan(4t - \frac{3\pi}{4}) \)

This expression directly shows how \( x \) changes with \( t \). Explicit solutions are often preferred in many applications because they give a clear functional relationship of one variable in terms of another, making it easier to compute and analyze specific values.

Differential equations are mathematical equations that involve functions and their derivatives. They are used extensively in fields ranging from physics to finance to describe how things change. Solving differential equations often involves finding a function that satisfies the given equation.

The given problem is a first-order differential equation:\

• \( \frac{dx}{dt} = 4(x^2 + 1) \)

Here, \( \frac{dx}{dt} \) represents the rate of change of \( x \) with respect to \( t \). This equation tells us that the rate of change depends on \( x^2 + 1 \).

To solve such equations, techniques like separating variables are often used, allowing us to integrate each side separately.

• Separation of variables: Transform the equation into a form where all terms involving \( x \) are on one side and all terms involving \( t \) are on the other.

Understanding differential equations and their solutions, both implicit and explicit, is crucial for modeling and solving real-world problems where change over time is an important aspect.