Chapter 2: Q9E (page 76)

Use the method discussed under “Homogeneous Equations” to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$

Short Answer

Expert verified

Homogeneous equation for the given equation is $y = \frac{- x}{\ln | x | + C}$ and $y \equiv 0$ .

Step by step solution

General form of Homogeneous equation

If the right-hand side of the equation $\frac{dy}{dx} = f (x, y)$ can be expressed as a function of the ratio $\frac{y}{x}$ alone, then we say the equation is homogeneous.

Evaluate the given equation

Given, $(xy + y^{2}) dx - x^{2} dy = 0$ .

By Evaluating,

role="math" localid="1654867974528" $\begin{array}{rcl} (xy + y^{2}) dx - x^{2} dy & = & 0 \\ - x^{2} dy & = & - (xy + y^{2}) dx \\ \frac{dy}{dx} & = & \frac{(xy + y^{2})}{x^{2}} \\ = & \frac{y}{x} + \frac{y^{2}}{x^{2}} \\ = & \frac{y}{x} + {(\frac{y}{x})}^{2} \end{array}$

Substitution method

Let us take $v = \frac{y}{x}$ .

Then $y = vx$ .

By Differentiating,

$\begin{array}{rcl} \frac{dy}{dx} & = & v + x \frac{dv}{dx} \\ v + v^{2} & = & v + x \frac{dv}{dx} \\ v^{2} & = & x \frac{dv}{dx} \\ \frac{1}{v^{2}} dv & = & \frac{1}{x} dx \end{array}$

Now integrating on both sides,

role="math" localid="1654868488187" $\begin{array}{rcl} \int v^{- 2} dv & = & \int \frac{1}{x} dx \\ - v^{- 1} & = & \ln |x| + C \\ - \frac{1}{v} & = & \ln |x| + C \end{array}$

Substitute $v = \frac{y}{x}$ .

$\begin{array}{rcl} - \frac{1}{\frac{y}{x}} & = & \ln |x| + C \\ - \frac{x}{y} & = & \ln |x| + C \\ y & = & - \frac{x}{\ln |x| + C}, y \equiv 0 \end{array}$

Therefore, Homogeneous equation for the given equation is $y = \frac{- x}{\ln | x | + C}$ and $y \equiv 0$ .

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Use the method discussed under “Homogeneous Equations” to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$

Short Answer

Step by step solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Use the method discussed under “Homogeneous Equations” to solve problems 9 - 16. (xy+y2)dx-x2dy=0

Short Answer

Step by step solution

General form of Homogeneous equation

Evaluate the given equation

Substitution method

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Pure Maths

Statistics

Mechanics Maths

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the method discussed under “Homogeneous Equations” to solve problems 9 - 16.
$(xy + y^{2}) dx - x^{2} dy = 0$