Question

In Exercises \(11-14\) , find the solution of the differential equation \(d y / d t=k y, k\) a constant, that satisfies the given conditions. $$k=-0.5, \quad y(0)=200$$

Short answer

The solution of the differential equation \(\frac{dy}{dt}=-0.5y\) given the initial condition \(y(0) = 200\) is \(y(t) = 200e^{-0.5t}\)

Step-by-step solution

Write the Differential Equation

The differential equation is given as \(\frac{dy}{dt}=-0.5y\).

Find the General Solution

This is a first order linear differential equation and is separable. So, separate the variables to give \(\frac{1}{y} dy = -0.5 dt\). Integrate both sides to get the general solution. The integration of both side results in \(ln|y| = -0.5t + C\) where \(C\) is the constant of integration.

Find the constant of integration \(C\)

To find the value of the \(C\), substitute the initial condition \(y(0)=200\) in the general solution. Therefore, \(ln|200| = -0.5*0 + C\) thus, \(C= ln(200)\).

Insert C into the General Solution

Re-write the general solution by inserting the value of \(C\): \(ln|y| = -0.5t + ln(200)\). Lastly, reformat the equation to present solution in a clearer way, this results in \(y(t) = 200e^{-0.5t}\).