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Expert-verified Found in: Page 1 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # In Problems 23-28, determine whether Theorem 1 implies that the given initial value problem has a unique solution.$\frac{\mathbf{dy}}{\mathbf{dx}}{\mathbf{=}}{{\mathbf{y}}}^{{\mathbf{4}}}{\mathbf{-}}{{\mathbf{x}}}^{{\mathbf{4}}}{\mathbf{,}}{\mathbf{}}{\mathbf{y}}\left(0\right){\mathbf{=}}{\mathbf{7}}$

The hypotheses of Theorem 1 are satisfied.

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

See the step by step solution

## Step 1: 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 $\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=4{\mathrm{y}}^{3}$.

## Step 2: 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.

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