For the differential equation \( d y / d t+5 \dot{y}=0 \) with \( y(0)=1 \), the general solution is

To solve the differential equation \( \frac{d y}{d t} + 5y = 0 \), we can identify it as a first-order linear ordinary differential equation. Rewriting it gives us \( \frac{d y}{d t} = -5y \). This can be solved using separation of variables. Rearranging yields \( \frac{dy}{y} = -5 dt \). Integrating both sides, we have \( \ln|y| = -5t + C \), or equivalently, \( y = Ce^{-5t} \). Using the initial condition \( y(0) = 1 \), we find \( C = 1 \). Thus, the particular solution is \( y(t) = e^{-5t} \). So, the final answer is: \[ y(t) = e^{-5t}. \] Now, let’s explore a bit more about our differential equation! Before differential equations were formalized, mathematicians like Newton and Leibniz were laying the groundwork for calculus in the 17th century. These foundational ideas eventually led to the development of differential equations as a powerful tool for modeling real-world phenomena, such as population growth and the motion of objects! When solving first-order linear differential equations, a common mistake is forgetting to apply initial conditions, which leads to a general solution without a specific function. Make sure to substitute any given initial values to find your particular solution! It's crucial for checksum validation in real-world applications, improving accuracy and reliability in your results!