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The term "'homogeneous'" is used in more than one context in mathematics. Perhaps the most prominent are the following three distinct cases:

1. Homogeneous functions
2. Homogeneous type of first order differential equations
3. Homogeneous differential equations (in contrast to "inhomogeneous" differential equations). This definition is used to define a property of certain linear differential equations—it is unrelated to the above two cases.

Each one of these cases will be briefly explained as follows.

Homogeneous functions

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Definition. A function  $f(x)$  is said to be homogeneous of degree   $n$   if, by introducing a constant parameter  $\lambda$, replacing the variable   $x$   with   $\lambda x$   we find:

$f(\lambda x)={\lambda }^{n}f(x).$

This definition can be generalized to functions of more-than-one variables; for example, a function of two variables $f(x,y)$ is said to be homogeneous of degree  $n$  if we replace both variables  $x$  and  $y$  by  $\lambda x$  and  $\lambda y$,  we find:

$f(\lambda x,\lambda y)={\lambda }^{n}f(x,y).$

Example. The function  $f(x,y)=(2x^2-3y^2+4xy)$  is a homogeneous function of degree 2 because:

$f(\lambda x,\lambda y)=[2(\lambda x{)}^2-3(\lambda y{)}^2+4(\lambda x\lambda y)]=(2{\lambda }^2{x}^2-3{\lambda }^2{y}^2+4{\lambda }^{2}xy)={\lambda }^2(2x^2-3y^2+4xy)={\lambda }^{2}f(x,y).$

This definition of homogeneous functions has been used to classify certain types of first order differential equations.

Homogeneous type of first-order differential equations

A first-order ordinary differential equation in the form:

$M(x,y)dx+N(x,y)dy=0$

is a homogeneous type if both functions M(x, y) and N(x, y) are homogeneous functions of the same degree n. That is, multiplying each variable by a parameter  $\lambda$, we find:

$M(\lambda x,\lambda y)={\lambda }^{n}M(x,y)$     and     $N(\lambda x,\lambda y)={\lambda }^{n}N(x,y).$

Thus,

$\frac{M(\lambda x,\lambda y)}{N(\lambda x,\lambda y)}=\frac{M(x,y)}{N(x,y)}.$

Solution method

In the quotient   $\frac{M(tx,ty)}{N(tx,ty)}=\frac{M(x,y)}{N(x,y)}$, we can let   $t=1/x$   to simplify this quotient to a function $f$ of the single variable $y/x$:

$\frac{M(x,y)}{N(x,y)}=\frac{M(tx,ty)}{N(tx,ty)}=\frac{M(1,y/x)}{N(1,y/x)}=f(y/x).$

Introduce the change of variables $y=ux$; differentiate using the product rule:

$\frac{d(ux)}{dx}=x\frac{du}{dx}+u\frac{dx}{dx}=x\frac{du}{dx}+u,$

thus transforming the original differential equation into the separable form:

$x\frac{du}{dx}=f(u)-u;$

this form can now be integrated directly (see ordinary differential equation).

Special case

A first order differential equation of the form (a, b, c, e, f, g are all constants):

$(ax+by+c)dx+(ex+fy+g)dy=0,$

can be transformed into a homogeneous type by a linear transformation of both variables ($\alpha$ and $\beta$ are constants):

$t=x+\alpha ;z=y+\beta .$

Homogeneous linear differential equations

Definition. A linear differential equation is called homogeneous if the following condition is satisfied: If  $\varphi (x)$  is a solution, so is  $c\varphi (x)$, where $c$ is an arbitrary (non-zero) constant. Note that in order for this condition to hold, each term in a linear differential equation of the dependent variable y must contain y or any derivative of y; a constant term breaks homogeneity. A linear differential equation that fails this condition is called inhomogeneous.

A linear differential equation can be represented as a linear operator acting on y(x) where x is usually the independent variable and y is the dependent variable. Therefore, the general form of a linear homogeneous differential equation is of the form:

$L(y)=0$

${\displaystyle }$where L is a differential operator, a sum of derivatives, each multiplied by a function  $f_i$  of x:

$L=\sum _{i=1}^n{f}_i(x)\frac{d^i}{d{x}^i};$

where  $f_i$  may be constants, but not all  $f_i$  may be zero.

For example, the following differential equation is homogeneous

$\sin (x)\frac{d^{2}y}{d{x}^2}+4\frac{dy}{dx}+y=0,$

whereas the following two are inhomogeneous:

$2x^2\frac{d^{2}y}{d{x}^2}+4x\frac{dy}{dx}+y=\cos (x);$

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

See also

• Method of separation of variables

Notes

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References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. (This is a good introductory reference on differential equations.)
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010. (This is a classic reference on ODEs, first published in 1926.)

External links

• Homogeneous differential equations at MathWorld
• Wikibooks: Ordinary Differential Equations/Substitution 1