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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-5E (page 46)

In problem $1 - 6$ , determine whether the differential equation is separablerole="math" localid="1654775979001" $(x y^{2} + 3 y^{2}) d y - 2 x d x = 0$ .

Short Answer

Expert verified

The differential equation $(x y^{2} + 3 y^{2}) d y - 2 x d x = 0$ 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 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

$(x y^{2} + 3 y^{2}) d y - 2 x d x = 0 . . . . . . . . . . (1)$

Equation (i) can be written as

$y^{2} (x + 3) d y - 2 x d x = 0 \frac{d y}{d x} = \frac{2 x}{y^{2} (x + 3)}$

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

$f (x, y) = \frac{2 x}{y^{2} (x + 3)} = \frac{2 x}{x^{2} + 3} \frac{1}{y^{2}}$

This function can be written as a product of two functions $g (x)$ and $h (y)$ defined as,

$g (x) = \frac{2 x}{x^{2} + 3} = \frac{1}{y^{2}}$

Therefore, the given differential equation is separable.

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Most popular questions from this chapter

In problems, 1-8 identify (do not solve) the equation as homogeneous, Bernoulli, linear coefficients, or of the form $y^{'} = G (a x + b y)$ . $(t + x + 2) d x + (3 t - x - 6) d t = 0$

Question: In problems 33-40 Solve the equation given in Problem 3.

$\frac{dy}{dx} + \frac{y}{x} = x^{3} y^{2}$

In problem $1 - 6$ , determine whether the differential equation is separable $\frac{d y}{d x} = \frac{y e^{x + y}}{x^{2} + 2}$ .

Question: In Problems 1 - 30, solve the equation.

$鈥� (y^{3} + 4 e^{x} y) dx + (2 e^{x} + 3 y^{2}) dy = 0$

Question: In Problems 31-40, solve the initial value problem.

$(2 y^{2} + 4 x^{2}) dx - xydy = 0, 鈥夆赌� y (1) = - 2$

See all solutions

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