Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), find y.
Answer Details
Given that \(\frac{\mathrm d y}{\mathrm d x} = \sqrt{x}\), we can integrate both sides with respect to x to obtain the original function y(x).
$$
\frac{\mathrm d y}{\mathrm d x} = \sqrt{x} \\
\int \frac{\mathrm d y}{\mathrm d x} \mathrm d x= \int \sqrt{x} \mathrm d x\\
y= \int x^{\frac{1}{2}} \mathrm d x \\
y= \frac{2}{3}x^{\frac{3}{2}} + c
$$
where c is an arbitrary constant of integration. Therefore, the correct option is (b) \(\frac{2}{3}x^{\frac{3}{2}} + c\).