Question

The initial value problem \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{y}}^{\frac{{\bf{2}}}{{\bf{3}}}}}{\bf{,}}\;{\bf{y(2) = 1}}{{\bf{0}}^{{\bf{ - 100}}}}\]has a unique solution in some open interval around x = 2.

Short answer

The given statement is true.

Step-by-step solution

Finding partial derivatives

Since\[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{y}}^{\frac{{\bf{2}}}{{\bf{3}}}}}\]

Then\[\frac{{\partial {\bf{f}}}}{{\partial {\bf{y}}}}{\bf{ = }}\frac{{\bf{2}}}{{\sqrt[{\bf{3}}]{{\bf{y}}}}}\]

Checking the final result

The given function is continuous at x = 2.So the function is continuous at some open interval x = 2.

Therefore, the statement is true.