Q23 E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

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

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

Step-by-Step Solution

Answer

The hypotheses of Theorem 1 are satisfied. 

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