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

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 $f (x, y) = g (x) + h (y)$ .

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 – 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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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

Step by step solution

Concept of Separable Differential Equation

Determining whether the equation is Separable or not

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In problem 1-6, determine whether the differential equation is separablerole="math" localid="1654775979001" (xy2+3y2)dy-2xdx=0.

Short Answer

Step by step solution

Concept of Separable Differential Equation

Determining whether the equation is Separable or not

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Study anywhere. Anytime. Across all devices.

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