Solving a Nonlinear Differential Equation with Initial Condition

Solve the differential equation $\frac{du}{dt} = 2t + \frac{\sec^2 t}{2u}$ with the initial condition $u(0) = -5$.

Differential equations are equations that involve a function and its derivatives, and they are fundamental to modeling various phenomena in engineering, physics, economics, and other fields. One of the key strategies in solving differential equations is understanding whether the equation is linear or nonlinear, as this greatly influences the method used to find the solution. In this problem, we encounter a first-order differential equation that is nonlinear due to the presence of the square of the secant function and its dependence on both the independent variable, t, and the dependent variable, u.

To tackle this equation, we must utilize techniques suitable for separable equations, which allow us to rewrite the differential equation in a way that separates the variables u and t on different sides of the equation. This method transforms the problem into one of finding antiderivatives, which is often simpler than dealing with the equation in its original form. The presence of the initial condition, $u(0) = -5$, is critical for determining the particular solution to the differential equation, as it allows us to solve for any constants of integration that arise from the antiderivatives.

Understanding the concepts of initial conditions and the method of separation of variables is crucial, as these tools are widely applied not just in this specific problem but across a variety of situations involving differential equations. They also highlight the importance of checking for possible closed-form antiderivatives and handling nonlinear terms that complicate the process. The ability to manipulate and solve such equations opens doors to analyzing real-world problems where dynamic relationships are expressed as differential equations.