## Initial Value Problems and Solutions**Problem 3:** For each of the following initial value problems, write down a formula for the solution $ x(t) $ and determine its maximal interval of existence.### (a)\[ \dot{x} = x^2 - 4 \quad \text{with} \quad x(0) = 0 \]### (b)\[ \dot{x}_1 = x_1^2, \quad \dot{x}_2 = x_2 + x_1^{-1} \quad \text{with} \quad x_1(0) = 1, \quad x_2(0) = 1 \] In this problem, the objective is to find solutions $ x(t) $ based on the given initial conditions and determine the interval over which these solutions are valid.

Answered: For each of the following initial value…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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## Initial Value Problems and Solutions

**Problem 3:** For each of the following initial value problems, write down a formula for the solution $ x(t) $ and determine its maximal interval of existence.

### (a)
\[ \dot{x} = x^2 - 4 \quad \text{with} \quad x(0) = 0 \]

### (b)
\[ \dot{x}_1 = x_1^2, \quad \dot{x}_2 = x_2 + x_1^{-1} \quad \text{with} \quad x_1(0) = 1, \quad x_2(0) = 1 \]

In this problem, the objective is to find solutions $ x(t) $ based on the given initial conditions and determine the interval over which these solutions are valid.

Expert Solution

Step 1

Hi! As per norm, we will be solving only the first question. If you need an answer to the other then kindly repost the question by specifying it.

For (a),

We need to solve $\dot{x} = x^{2} - 4$ with $x (0) = 0$ .

So,

$\dot{x} = x^{2} - 4 \Rightarrow \frac{d x}{d t} = x^{2} - 4 \Rightarrow \frac{1}{x^{2} - 4} d x = d t Integrating both sides we get, \Rightarrow \int \frac{1}{x^{2} - 4} d x = \int d t \Rightarrow - \frac{1}{4} (\ln (\frac{x}{2} + 1) - \ln (\frac{x}{2} - 1)) = t + c_{1}$