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Q26 E

Expert-verifiedFound in: Page 14

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.**

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

The hypotheses of Theorem 1 are satisfied.

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

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

and

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

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.**

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