Question

True or False The general solution to \(d y / d t=2 y\) can be written in the form \(y=C\left(3^{k t}\right)\) for sor some constants \(C\) and \(k .\) Justify your answer.

Short answer

False, the general solution to \(d y / d t=2 y\) cannot be written in the form \(y=C\left(3^{k t}\right)\) for some constants \(C\) and \(k\).

Step-by-step solution

- Identify the type of differential equation

The differential equation presented, \(d y / d t=2 y\), is a first order homogeneous linear differential equation.

- Solve the Differential Equation

The solution to this type is given by the formula \(y(t) = Ce^{kt}\), where \(C\) and \(k\) are constants. Here, \(k = 2\). So, the solution of the given differential equation is \(y = Ce^{2t}\)

- Compare the solutions

The solution provided in the question is \(y=C\left(3^{k t}\right)\). Now, we notice that this given function can only match our differential equation solution if the base of the exponent (3 in this case) equals to \(e^2\). However, this is not possible because \(e^2\) is approximately 7.39, not 3.