Question

(a) Give the domain of the function$\mathit{y}\mathbf{=}{\mathit{x}}^{\frac{2}{3}}$.

(b) Give the largest interval I of definition over which$\mathit{y}\mathbf{=}{\mathit{x}}^{\frac{2}{3}}$is a solution of the differential equation$\mathbf{3}\mathit{x}\mathit{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathit{y}\mathbf{=}\mathbf{0}$.

Short answer

(a) The domain of the function is $\mathrm{ℝ}$.

(b) The largest interval for the solution $y=x^{\frac{2}{3}}$is and .

Step-by-step solution

Define the domain and the interval of the function.

The domain of a function in mathematics is the set of inputs that the function accepts. The term "interval notation" refers to a method of representing a subset of real numbers using the numbers that connect them.

The sensitivity of the function is derived by the derivative of a function of a real variable for modifying the argument.

Step 2(a): Determine the domain

As $\frac{2}{3}>0$, then the function $y=x^{\frac{2}{3}}$is defined for all $x\in R$.

Hence, the domain is a set of all real numbers.

Step 3(b): Determine the largest interval

Let the function defined and differentiable on the largest intervalbe y(x).

The derivative of the function y(x) is given by,

$\frac{\partial y}{\partial x}=\frac{2}{3}{x}^{-\frac{1}{3}}$

As the function is not defined for $x=0$, the largest interval for the solution are and .

Hence, the largest interval is either or .