Question

(a) Using the separate-and-integrate method, find the general solution to the differential equation: dx/dt = 1/3x. Show all of your work and the steps involved in finding the general solution.(b) Find the particular solution with the initial condition x(0)=4.

Answer 1

To solve the differential equation dx/dt = 1/3x using the separate-and-integrate method, separate the variables and integrate both sides. The general solution is given by x = ±4e^((1/3)t + ln(4)).

Step-by-step explanation:

To solve the differential equation dx/dt = 1/3x using the separate-and-integrate method, we can start by separating the variables and integrating both sides. Rearranging the equation, we have dx/x = (1/3)dt. Integrating both sides, we get ln|x| = (1/3)t + C, where C is the constant of integration.

To find the general solution, we exponentiate both sides to eliminate the natural logarithm. This gives us |x| = e^((1/3)t + C).

For the particular solution with the initial condition x(0) = 4, we substitute t = 0 and x = 4 into the general solution to find the value of the constant of integration. In this case, we have 4 = e^(C), which implies that C = ln(4). Substituting this value back into the general solution gives x = ±4e^((1/3)t + ln(4)).