Chapter 6: Q1E (page 326)

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.
$xy''' - 3 y' + e^{x} y = x^{2} - 1 y (- 2) = 1, y' (- 2) = 0, y'' (- 2) = 2$

Short Answer

Expert verified

Thus, the largest interval is $(- \infty, 0)$ .

Step by step solution

Solve the given equation

The given equation is $xy''' - 3 y' + e^{x} y = x^{2} - 1 .$

Divide both sides by x in the above equation,

$y''' - 3 (\frac{1}{x}) y' + e^{x} (\frac{1}{x}) y = x^{2} - 1 (\frac{1}{x})$

Compare with the standard form of a linear differential equation,

$y''' + p (x) y'' + q (x) y' + r (x) y = s (x)$

Therefore,

$q (x) = \frac{- 3}{x}, r (x) = \frac{e^{x}}{x}, s (x) = \frac{x^{2} - 1}{x}$

Step 2: Check the continuity

$q (x) = \frac{- 3}{x}$ is continuous whenever $x \neq 0 .$

$r (x) = \frac{e^{x}}{x}$ is continuous whenever $x \neq 0 .$ and

$s (x) = \frac{x^{2} - 1}{x}$ is continuous whenever $x \neq 0 .$

The largest interval (a, b)

Now overall q, r, and s are continuous in $\forall x \in (- \infty, 0) \cup (0, \infty) .$

The initial condition is defined as $x_{0} = - 2$ and $- 2 \in (- \infty, 0) .$

Hence, the largest interval $(- \infty, 0) .$

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Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.
$xy''' - 3 y' + e^{x} y = x^{2} - 1 y (- 2) = 1, y' (- 2) = 0, y'' (- 2) = 2$

Short Answer

Step by step solution

Solve the given equation

Step 2: Check the continuity

The largest interval (a, b)

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Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.xy'''-3y'+exy=x2-1y-2=1, y'-2=0, y''-2=2

Short Answer

Step by step solution

Solve the given equation

Step 2: Check the continuity

The largest interval (a, b)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Probability and Statistics

Discrete Mathematics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

Determine the largest interval (a, b) for which Theorem 1 guarantees the existence of a unique solution on (a, b) to the given initial value problem.
$xy''' - 3 y' + e^{x} y = x^{2} - 1 y (- 2) = 1, y' (- 2) = 0, y'' (- 2) = 2$