Question

Find the function \(y=f(t)\) passing through the point (0,10) with the given first derivative. Use a graphing utility to graph the solution. $$ \frac{d y}{d t}=-\frac{1}{2} y $$

Short answer

The particular solution to the differential equation is \(y = 10 e^{-\frac{1}{2} t}\).

Step-by-step solution

Understand the given information

The given equation is a simple differential equation. The derivative of \(y\) with respect to \(t\) is \(-\frac{1}{2} y\). The goal is to find \(y\) as a function of \(t\) (also denoted as \(y=f(t)\)). The function \(y=f(t)\) passes through the point (0,10).

Solve the differential equation

Firstly, rearrange \(-\frac{1}{2} y = \frac{d y}{d t}\) to \(-\frac{2}{y} dy = dt\. integrating both sides get \(-2 \ln |y| = t + C_1\), where \(C_1\) is the constant of integration. Apply the exponential function to both sides to remove the natural logarithm. The result is \(y = e^{-\frac{1}{2} t + C_1}\) or \(y = e^{C_1} e^{-\frac{1}{2} t}\). By observing that \(e^{C_1}\) is just a constant itself, which can be denoted by C, the general solution to the differential equation is \(y = C e^{-\frac{1}{2} t}\).

Find the particular solution

To find the constant \(C\), substitute the point (0,10) into the general solution. Thus \(10 = C e^{-\frac{1}{2} \cdot 0}\), which simplifies to \(10 = C e^0\), then \(10 = C\). So the particular solution to the differential equation is \(y = 10 e^{-\frac{1}{2} t}\).

Graph the function

The final step is graphically representing the function. Use a graphing utility to produce a graph for \(y = 10 e^{-\frac{1}{2} t}\). The graph should start at (0,10) and decay exponentially as \(t\) increases, reaching asymptotically towards the \(t\)-axis.