Trig Substitution with Initial Condition Problem

dy/dt = $\frac{1}{25 + 9t^2}$, with initial condition $y\left(\frac{5}{3}\right) = \frac{\pi}{30}$.

This problem presents a first-order ordinary differential equation (ODE) with an initial condition. The first step in solving such a problem involves recognizing the form of the differential equation and identifying whether it can be solved using analytical methods. Here, the differential equation provided is separable, meaning that we can rearrange it to express each variable and its differential separately on each side of the equation. This technique takes advantage of integral calculus to solve for the function y for given values of t.

Begin by separating the variables, which will allow the integration of each side independently. This involves algebraic manipulation to isolate dy in terms of y and dt in terms of t. Once the integral form is obtained, the problem reduces to performing the integration over the specified limits. Integrals in problems like these often lead to discussion of techniques for integration, and may sometimes involve partial fraction decomposition, trigonometric identities, or straightforward polynomial integration, depending on the complexity of the expression involved.

After integration, the use of the initial condition is crucial. The initial condition is a specific value that is used to determine the constant of integration—a term that appears when indefinite integrals are evaluated. By substituting the initial condition into the integrated equation, you can solve for this constant. Thus, the problem-solving process exemplified in this problem involves both algebraic manipulation, calculus techniques for integration, and application of initial conditions to find a specific solution to the ODE.

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