Use the given general solution to find a solution of the...

Use the given general solution to find a solution of the differential equation having the given initial condition. Sketch the solution, the initial condition, and discuss the solution's interval of existence. $$ \begin{aligned} &y^{\prime}+4 y=\cos t, y(t)=(4 / 17) \cos t+(1 / 17) \sin t+C e^{-4 t}, \\ &y(0)=-1 \end{aligned} $$

Use the given general solution to find a solution of the differential equation having the given initial condition. Sketch the solution, the initial condition, and discuss the solution's interval of existence.
$$
\begin{aligned}
&y^{\prime}+4 y=\cos t, y(t)=(4 / 17) \cos t+(1 / 17) \sin t+C e^{-4 t}, \\
&y(0)=-1
\end{aligned}
$$

Key Concepts

Interval of Existence

The interval of existence refers to the range of the independent variable over which the solution is valid and unique. For first-order linear differential equations with continuous coefficients, the Existence and Uniqueness Theorem guarantees that the solution exists and is unique over any interval where the coefficient functions remain continuous. This concept highlights the importance of analyzing the behavior of the functions involved to determine the proper domain for the solution.

Initial Conditions and Determining Constants

Initial conditions specify the value of the solution at a particular point, which is used to determine the arbitrary constant present in the general solution. This process yields a unique particular solution that satisfies both the differential equation and the initial condition, ensuring the result is tailored to the problem at hand.

General and Particular Solutions

The general solution of a differential equation combines the particular solution, which is a specific solution to the nonhomogeneous equation, and a complementary solution, which is the general solution to the homogeneous part of the equation. Understanding this separation is crucial, as it allows one to capture all possible solutions of the differential equation by introducing an arbitrary constant.

First-Order Linear Differential Equations

These equations have the form y' + p(t)y = g(t). The method involves using an integrating factor to combine the derivative and the coefficient of y, allowing the left-hand side to be written as the derivative of a product. This technique transforms the original equation into one that is directly integrable, forming the foundation for finding its general solution.

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