Q24 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.

$\frac{\mathbf{dy}}{\mathbf{dt}}\mathbf{-}\mathbf{ty}\mathbf{=}{\mathbf{sin}}^{\mathbf{2}}\mathbf{t}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{y}\left(\pi \right)\mathbf{=}\mathbf{5}$

Step-by-Step Solution

Answer

The hypotheses of Theorem 1 are satisfied. 

It follows from the theorem that the given initial value problem has a unique solution.

Step 1: Finding the partial derivative of the given relation concerning y.

Here, $\mathrm{f}\left(\mathrm{t},\mathrm{y}\right)={\sin }^2 \mathrm{t}+\mathrm{ty}$ and $\frac{\partial \mathrm{f}}{\partial \mathrm{y}}=\mathrm{t}$.

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 $\mathrm{f}\left(t,\mathrm{y}\right)$ and $\frac{\partial \mathrm{f}}{\partial \mathrm{y}}$ are continuous in any rectangle containing the point $\left(\pi ,5\right)$, so the hypotheses of Theorem 1 are satisfied. It then follows from the theorem that the given initial value problem has a unique solution in an interval about $\mathrm{t}=\pi$ of the form $\left(\pi -\delta , \pi +\delta \right)$, where is some positive number.

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