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Fundamentals Of Differential Equations And Boundary Value Problems

R. Kent Nagle, Edward B. Saff, Arthur David Snider

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 616 pages

ISBN: 9780321977069

Chapter 2: Q-3E (page 46)

In problem $1 - 6$ , determine whether the differential equation is separable $\frac{ds}{dt} = t l n (s^{2 t}) + 8 t^{2}$ .

Short Answer

Expert verified

The differential equation $\frac{d s}{d t} = t \ln (s^{2 t}) + 8 t^{2}$ is separable.

Step by step solution

01

Concept of Separable Differential Equation

A first-order ordinary differential equation $\frac{d y}{d x} = f (x, y)$ is referred to as separable if the function in the right-hand side of the equation is expressed as a product of two functions $g (x)$ that is a function of $x$ alone and $h (y)$ that is a function of $y$ alone.

Mathematically, the equation $\frac{d y}{d x} = f (x, y)$ is separable, when $f (x, y) = g (x) + h (y)$ .

02

Determining whether the equation is Separable or not

The given equation is

$\frac{d s}{d t} = t \ln (s^{2 t}) + 8 t^{2} . . . . . . . . (1)$

The function in the right-hand side of equation (1) is

$f (t, s) = t \ln (s^{2 t}) + 8 t^{2} = t (2 t l n s) + 8 t^{2} = 2 t^{2} \ln s b + 8 t^{2} = 2 t^{2} \ln s + 8 t^{2} = 2 t^{2} (\ln s + 4)$

This function can be written as a product of two functions and defined as,

$\begin{array}{l} g (t) = 2 t^{2} \\ h (s) = \ln s + 1 \end{array}$

Therefore, the given differential equation is separable.

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