Q25 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.

$\mathbf{3}\mathbf{x}\frac{\mathbf{dx}}{\mathbf{dt}}\mathbf{+}\mathbf{4}\mathbf{t}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{ }\mathbf{x}\left(2\right)\mathbf{=}\mathbf{-}\mathbf{\pi }$

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, $\mathrm{f}\left(\mathrm{t},\mathrm{x}\right)=-\frac{4\mathrm{t}}{3\mathrm{x}}$ and

$\begin{array}{l}\frac{\partial \mathrm{f}}{\partial \mathrm{x}}=-12\mathrm{t}\times \left(-1\right)\times {\mathrm{x}}^{-2}\\ \frac{\partial \mathrm{f}}{\partial \mathrm{x}}=\frac{12\mathrm{t}}{{\mathrm{x}}^2}\end{array}$

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 $\mathrm{f}\left(t,x\right)$ and $\frac{\partial \mathrm{f}}{\partial x}$ are continuous in any rectangle containing the point $\left(2,-\pi \right)$, 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 $\mathrm{t}=2$ of the form $\left(2-\delta ,2+\delta \right)$, where $\delta$ is some positive number.

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