Question

As in text just before (4.11), show that

(a) ${\mathit{x}}^{\mathbf{2}}\mathbf{-}\mathbf{5}\mathit{x}\mathit{y}\mathbf{+}{\mathit{y}}^{\mathbf{3}}\mathbf{/}\mathit{x}$is a homogeneous function of degree 2;

(b) ${\mathit{x}}^{\mathbf{-}\mathbf{1}}\left(y^4-x^{3}y\right)\mathbf{-}\mathit{x}{\mathit{y}}^{\mathbf{2}}\mathbf{sin}\left(x/y\right)$ is homogeneous of degree 3;

(c) ${\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}\mathbf{+}{\mathit{x}}^{\mathbf{5}}\mathit{I}\mathit{n}\left(y/x\right)\mathbf{-}{\mathit{y}}^{\mathbf{6}}\mathbf{/}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}$ is homogeneous of degree 5;

(d)${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathit{y}\mathbf{,}\mathit{x}\mathbf{+}\mathbf{cos}\mathbf{}\mathit{y}\mathbf{,}\mathbf{}\mathit{a}\mathit{n}\mathit{d}\mathbf{}\mathit{y}\mathbf{+}\mathbf{1}$,are not homogeneous.

See Chapter 4, Section 13, Problem 1 for a more general definition of a homogeneous function of any number of variables.

Short answer

a) The differential equation $x^2-2xy+\frac{y^3}{x}$is a homogeneous function of degree 2

b) $x^{-1}\left(y^4-x^{3}y\right)-x{y}^2\sin \left(\frac{x}{y}\right)$is a homogeneous function of degree 2.

c) $\left(x^2{y}^3+x^{5}In\left(\frac{y}{x}\right)\right)-\frac{y^6}{\sqrt{x^2+y^2}}$is a homogeneous function of degree 5

d) $x^2+y,x+\cos y,y+1$are not homogeneous function

Step-by-step solution

Given information

(a) $x^2-5xy+y^3/x$is a homogeneous function of degree 2;

(b) $x^{-1}(y^4-x^{3}y)-x{y}^2\sin (x/y)$is homogeneous of degree 3;

(c) role="math" localid="1665064171095" $x^2{y}^3+x^{5}In\left(y/x\right)-y^6/\sqrt{x^2+y^2}$is homogeneous of degree 5;

(d) $x^2+y,x+\cos y,andy+1$are not homogeneous

Homogenous function

Homogenous equation: A homogeneous function of x and y of degree nmeans a function can be written as ${\mathit{x}}^{\mathbf{n}}\mathit{f}\left(\frac{y}{x}\right)$

$\mathit{P}\left(x,y\right)\mathit{d}\mathit{x}\mathbf{+}\mathit{Q}\left(x,y\right)\mathit{d}\mathit{y}\mathbf{=}\mathbf{0}$

Where Pand Qare homogeneous function of the same degree is called homogeneous.

By the change of variable of homogeneous equation

role="math" localid="1665064922532" width="110" height="133">y'=dydx=-P(x,y)Q(x,y)=f(yx)

Replace by y = xvin equation.

Determine that x2-5xy+y3x is a homogeneous function of degree 2.

a)

By the change of variable of homogeneous equation

$\begin{array}{rcl}y\text{'}& =& \frac{\mathrm{dy}}{\mathrm{dx}}\\ & =& -\frac{\mathrm{P}(\mathrm{x},\mathrm{y})}{\mathrm{Q}(\mathrm{x},\mathrm{y})}\\ & =& f\left(\frac{\mathrm{y}}{\mathrm{x}}\right)\end{array}$

Replace by y = xv in equation

dy = xdv + vdx

Consider the equation$\begin{array}{rcl}& & x^2-5xy+\frac{y^3}{x}...............\left(1\right)\end{array}$

$\begin{array}{rcl}P(x,y)dx+Q(x,y)dy& =& 0\end{array}$

Where P and Q are homogeneous function of the same degree is called homogeneous

By the change of variable of homogeneous equation

$\begin{array}{rcl}y\text{'}& =& \frac{dy}{dx}\\ & =& -\frac{P\left(x,y\right)}{Q\left(x,y\right)}\\ & =& f\left(\frac{y}{x}\right)\end{array}$

Replace by y = xv in equation

Comparing the equation (1) with equation (2)

$\begin{array}{rcl}& & x^2-5xy+\frac{y^3}{x}\\ & & x^2-5xy+y^3{x}^{-1}\end{array}$

This is a homogeneous function of degree 2.

Determine that the differential equation x-1(y4 - x3y) - xy2 sin(x/y) is a homogeneous function of degree 31.

b)

By the change of variable of homogeneous equation

Replace by in equation

Consider the equation,

Where and are homogeneous function of the same degree is called homogeneous

By the change of variable of homogeneous equation

Replace by in equation

Comparing the equation (1) with equation (2)

This is a homogeneous function of degree 3