Question

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

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

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 concerning y.

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

and

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

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.