Q26 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.

$\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{cos}\mathbf{ }\mathbf{x}\mathbf{=}\mathbf{sin}\mathbf{ }\mathbf{t}\mathbf{,}\mathbf{ }\mathbf{x}\left(\pi \right)\mathbf{=}\mathbf{0}$

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 concerning y.

Here, $\mathrm{f}\left(\mathrm{t},\mathrm{x}\right)=\mathrm{sint}-\mathrm{cosx}$

and

$\begin{array}{l}\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=-\left(-\sin \mathrm{x}\right)\\ \frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\sin \mathrm{x}\end{array}$

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 $\mathrm{f}\left(\mathrm{t},\mathrm{x}\right)$ and $\frac{\partial \mathrm{f}}{\partial x}$ are continuous in any rectangle containing the point $\left(\pi ,0\right)$, 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 $\mathrm{t}=\pi$ about of the form $\left(\pi -\delta ,\pi +\delta \right)$, where $\delta$ is some positive number.

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