Question

In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}{\mathbf{y}}^{\mathbf{4}}\mathbf{-}{\mathbf{x}}^{\mathbf{4}}\mathbf{,}\mathbf{}\mathbf{y}\left(0\right)\mathbf{=}\mathbf{7}$

Short answer

The hypotheses of Theorem 1 are satisfied.

The theorem shows that the given initial value problem has a unique solution.

Step-by-step solution

Finding the partial derivative of the given relation with respect to y

Here, $\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)={\mathrm{y}}^4-{\mathrm{x}}^4$and$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=4{\mathrm{y}}^3$.

Determining whether Theorem 1 implies the existence of a unique solution or not.

Now from Step 1, we find that both of the functions$\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)$and$\frac{\partial \mathrm{f}}{\partial \mathrm{y}}$are continuous in any rectangle containing the point$\left(0,7\right)$, so the hypotheses of the Theorem are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an interval about$\mathrm{x}=0$of the form$\left(0-\delta ,0+\delta \right)$, where$\delta$is some positive number.

Hence, Theorem 1 implies that the given initial value problem has a unique solution.