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Fundamentals Of Differential Equations And Boundary Value Problems

Found in: Page 30

Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

The initial value problem \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{x}}^{\frac{{\bf{2}}}{{\bf{3}}}}}{\bf{,}}\;{\bf{x(0) = 1}}\] has a unique solution in some open interval around t = 0.

The given statement is true.

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Finding partial derivatives

Since \[\frac{{{\bf{dx}}}}{{{\bf{dt}}}}{\bf{ = 3}}{{\bf{x}}^{\frac{{\bf{2}}}{{\bf{3}}}}}\]

Then\[\frac{{\partial {\bf{f}}}}{{\partial {\bf{x}}}}{\bf{ = }}\frac{{\bf{2}}}{{\sqrt[{\bf{3}}]{{\bf{x}}}}}\]

Step 2: Checking the final result

Apply the initial conditions\[x\left( 0 \right) = 1\]

\[\frac{{\partial f}}{{\partial x}}{\bf{ = }}\frac{{\bf{2}}}{{\bf{0}}}\]

The result is infinite.

The given function is discontinuous a x = 0. So the function is not continuous in a rectangle containing the point (0,1).

Therefore, the statement is True.

Most popular questions for Math Textbooks

In Problems $15 - 24$ , solve for $Y (s)$ , the Laplace transform of the solution $y (t)$ to the given initial value problem.

$y'' + 3 y = t^{3}; y (0) = 0, y' (0) = 0$

Question: Let f(x) and g(x) be analytic at x₀. Determine whether the following statements are always true or sometimes false:

(a) 3f(x)+g(x) is analytic at x₀ .

(b) f(x)/g(x) is analytic at x₀ .

(c) f'(x) is analytic at x₀ .

(d) ${[f (x)]}^{3} - \int_{x_{0}}^{x} g (t) d t$ is analytic at x₀ .

In Problems 10–13, use the vectorized Euler method with h = 0.25 to find an approximation for the solution to the given initial value problem on the specified interval.

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Decide whether the statement made is True or False. The function $x (t) = t^{- 3} \sin t + 4 t^{- 3}$ is a solution to $t^{3} \frac{dx}{dt} = \cos t - 3 t^{2} x$ .

Consider the differential equation $\frac{d y}{d x} = x + s i n y$

⦁ A solution curve passes through the point $(1, \frac{π}{2})$ . What is its slope at this point?

⦁ Argue that every solution curve is increasing for $x > 1$ .

⦁ Show that the second derivative of every solution satisfies $\frac{d^{2} y}{{dx}^{2}} = 1 + x \cos y + \frac{1}{2} \sin 2 y .$

⦁ A solution curve passes through (0,0). Prove that this curve has a relative minimum at (0,0).

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