# 5.1 Диференцијални равенки од прв ред

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## 2. хомогена диференцијална равенка

Ако за функцијата $f\left(x,y\right)$ важи

$f\left(\text{tx},\text{ty}\right)={t}^{n}f\left(x,y\right)$ ,

таа се нарекува хомогена од n -ти ред.

Диференцијалната равенка од обликот

$M\left(x,y\right)\text{dx}+N\left(x,y\right)\text{dy}=0$

се нарекува хомогена диференцијална равенка ако функциите $M\left(x,y\right)$ и $N\left(x,y\right)$ се хомогени функции од ист ред. Хомогеноста на функциите кои се јавуваат во диференцијалната равенка овозможува таа да се запише во облик

$\frac{\text{dy}}{\text{dx}}=f\left(\frac{y}{x}\right)\text{.}$

Со воведување на смената

$\frac{y}{x}=z$

односно

$y=\text{zx}$

и со нејзино диференцирање се добива

${y}^{\text{'}}={z}^{\text{'}}x+z$

или

$\frac{\text{dy}}{\text{dx}}=x\frac{\text{dz}}{\text{dx}}+z$

и по заменување во хомогената диференцијална равнка, се добива диференцијална равенка од облик

${z}^{\text{'}}x=f\left(z\right)-z,$

што укажува дека променливите во вака добиената диференцијална равенка може да се раздвојат.

Притоа:

• Ако $f\left(z\right)-z\ne 0$ нејзиното решение е

$\int \frac{\text{dz}}{f\left(z\right)-z}=\int \frac{\text{dx}}{x}+\text{ln}C,$

• ако $f\left(z\right)=z$ нејзиното решение е

$\int \frac{\text{dy}}{y}=\int \frac{\text{dx}}{x}+\text{ln}C$

односно

$y=\text{Cx}\text{.}$

Пример 3.

Да се најде општото решение на диференцијалната равенка ${{y}^{2}+{x}^{2}{y}^{\text{'}}=\text{xy}y}^{\text{'}}\text{.}$

РЕШЕНИЕ.

Оваа диференцијална равенка е хомогена од втор ред бидејки ако ја решиме по изводот се добива

${y}^{\text{'}}=\frac{{y}^{2}}{\text{xy}-{x}^{2}},$

а по делење на изразот од десната страна (и броителот и именителот) со ${x}^{2}$ се добива

${y}^{\text{'}}=\frac{{\left(\frac{y}{x}\right)}^{2}}{\frac{y}{x}-1}\text{.}$

Со воведување на смената $y=\text{zx}$ и нејзиниот извод ${y}^{\text{'}}={z}^{\text{'}}x+z$ , таа се трансформира во диференцијална равенка од обликот

${z}^{\text{'}}x+z=\frac{{z}^{2}}{z-1},$

во која променливите се раздвојуваат

$\frac{z-1}{z}\text{dz}=\frac{1}{x}\text{dx}\text{.}$

Решението на оваа равенка се добива по интегрирање

$\int \frac{z-1}{z}\text{dz}=\int \frac{1}{x}\text{dx}+\text{ln}C$ ,

и тоа е

$z-\text{ln}\mid z\mid =\text{ln}\mid x\mid +\text{ln}C$ ,

односно

$z=\text{ln}\left(\text{Cxz}\right)$ .

Со враќање на старата променлива x преку смената $\frac{y}{x}=z$ , општото решение е

$\text{Cy}={e}^{\frac{y}{x}}\text{.}$

Пример 4.

Да се најде партикуларното решение на диференцијалната равенка

$\left(x{y}^{\text{'}}-y\right)\text{arctg}\frac{y}{x}=x,$ кое има вредност $y=0$ за $x=1\text{.}$

РЕШЕНИЕ:

Најпрво се бара општото решение на диференцијалната равенка. Таа е хомогена диференцијална равенка од прв ред бидејќи може да се запише во обликот

$\frac{\text{dy}}{\text{dx}}=\frac{1}{\text{arctg}\frac{y}{x}}+\frac{y}{x}\text{.}$

Со користење на смената $\frac{y}{x}=z$ , односно $y=\text{zx}$ и изводот ${y}^{\text{'}}={z}^{\text{'}}x+z$ ,

во диференцијалната рвенка променливите се раздвојуваат бидејќи

${z}^{\text{'}}x=\frac{1}{\text{arctg}z}\text{.}$

Оваа равенка има општо решение

$\int \text{arctg}z\text{dz}=\int \frac{\text{dx}}{x}+\text{ln}C,$

кое по решавање на интегралите е во облик

$z\text{arctg}z-\frac{1}{2}\text{ln}\left(1+{z}^{2}\right)=\text{ln}\text{Cx}$

или

$\frac{y}{x}\text{arctg}\frac{y}{x}-\frac{1}{2}\text{ln}\left(1+{\frac{y}{{x}^{2}}}^{2}\right)=\text{ln}\text{Cx}\text{.}$

Партикуларното решение ќе се добие со определување на константата C од почетните услови $y=0$ кога $x=1$ . Со замена на почетните услови во општото решение

$-\frac{1}{2}\text{ln}1=\text{ln}C⇒C=1$ .

Значи партикуларното решение е

$\frac{y}{x}\text{arctg}\frac{y}{x}-\frac{1}{2}\text{ln}\left(1+{\frac{y}{{x}^{2}}}^{2}\right)=\text{ln}x$

кое по антилогаритмирање е од обликот

$\sqrt{{x}^{2}+{y}^{2}}={e}^{\frac{y}{x}\text{arctg}\frac{y}{x}}$ . ◄

## 2.1. равенка која се сведува на хомогена диференцијална равенка

Ако диференцијалната равенка е од облик

${y}^{\text{'}}=f\left(\frac{\text{ax}+\text{by}+c}{{a}_{1}x+{b}_{1}y+{c}_{1}}\right),$

со смената

$\begin{array}{}x=u+\alpha ,\text{dx}=\text{du}\\ y=v+\beta ,\text{dy}=\text{dv}\end{array}$

каде $u,v$ се нови променливи а $\alpha ,\beta$ се константи, таа се сведува на

$\frac{\text{dv}}{\text{du}}=f\left(\frac{\text{au}+\text{bv}+\mathrm{a\alpha }+\mathrm{b\beta }+c}{{a}_{1}u+{b}_{1}v+{a}_{1}\alpha +{b}_{1}\beta +{c}_{1}}\right)$ .

Идејата за оваа смена е равенката да се сведе на хомогена и затоа константите $\alpha ,\beta$ се определуваат преку системот равенки

$\begin{array}{}\mathrm{a\alpha }+\mathrm{b\beta }+c=0\\ {a}_{1}\alpha +{b}_{1}\beta +{c}_{1}=0\text{.}\\ \\ \begin{array}{}\left\{\end{array}\\ \end{array}$

При тоа:

• Ако детерминантата на овој систем $\Delta ={\text{ab}}_{1}-{\text{ba}}_{1}\ne 0$ еднозначно се определуваат константите $\alpha ,\beta$ и равенката се сведува на хомогена диференцијална равенка

$\frac{\text{dv}}{\text{du}}=f\left(\frac{\text{au}+\text{bv}}{{a}_{1}u+{b}_{1}v}\right)$

која може да се реши со постапката за решавање на хомогена диференцијална равенка.

• Ако детерминантата на системот $\Delta ={\text{ab}}_{1}-{\text{ba}}_{1}=0$ со смената

Do i need a lab and a instructor to carry out Psychology empirical research?
Do i need a lab and a instructor to carry out Psychology empirical research?q
Nikhil
Karen
i forgot to add, its Social Psychology Research
Nikhil
Depends on whether it's clinical OR a naturalistic research
Wayne
I was here before but got lost
Karen
hello
Sheik
hi
California
hiiiii all
I wanna be a young psychologist....
Sheik
Define perception of human nature and its significance
Psychology is important for different programs to study because of its ability to build students critical thinking and by so doing, improve their ability to sieve information, thereby improving overall learning outcomes
I have a question about sex addiction how can you tell someone would have sex with anything like child man woman old young even an animal I have a feeling the guy I'm with is someone that could do anything because he has no self control and he is a narcissist that also has a addition to meth..
How did you get to meet this guy
Wayne
hello may i ask How long have you been seeing this guy and what kind of signs were givin to come to this conclusion,? along with what you've shared on top here?
Jo
Facebook but we had the same friends but never met before but have been together for 5 years and its been hell I now have PTSD and now no intrest in doing what I love because he puts me down for doing so or makes me feel stupid for feeling the way I do about things
Alexis
been together for 5 years things were good at first he was really nice if I needed something he made sure to get it but once I moved I with him he started calling me names but once we got our own place he got abusive and pretty much had me as a prisoner in my own home
Alexis
also when he gets high he gets very turned on to the point he would masturbate several times a day only time we would have sex is if I was asleep or we would fight to the point of abusing each other then he would want to have sex but he also seemed to be very weird around kids
Alexis
but if sober he wanted nothing to do with him he would actually make me take my brother to my mother's room and not let him come out and play with his toys also I had my son asleep in his room with the TV and night light on seth said he was going to get a shower well the water was on I thought he
Alexis
it sounds like this guys is someone who can be very dangerous if under the influence. I highly recommend getting yourself and children away from him. He may seem like a nice guy but in my limited experience its better to play it safe not only for your own safety but especially for the children.
April
you gotta leave the relationship.. it sounds like it won't end so well and the fact that he's using meth; that alone is a very huge red flag
GL
I agree with GL Rocketeer
April
yes I do also but I want to think it's just the drugs but he still is very one sided and not caring of others feelings
Alexis
one thing I could share with u from individuals ive spoken wit is that a person who smokes or especially runs it becomes extremely "turned on" and or to the point were they'll masturbate or seek outside their relationship to satisfy their wants or needs. so honestly i dont think its a bad I idea...
Jo
fr them in this situation out of his sight,out of his mind!-
Jo
so yea id agree, pick up and flee! Good luck in your situation and may God protect u and yours during this time!
Jo
thanks you guys I've asked for advice from family but no one actually listened to me and just pushed it aside
Alexis
I also agree wit miss april, about methin around wit this guy, and already abused u already kno wat ones capable of. theroriticly if one day he looses his sack or been up for few days to were sleep deprivation begins to alter his jugment and emotions , possibilitys Are high, and could hurt someone
Jo
it is better to make a plan, get everything ready. Then make your move, if it is drug out over time and this guy still has access to you. He will try to convince you he will change and stop. He will agree with everything you say to make you believe him. it will be lies.
gestalt theory of learning with example sultan
Rohit
A friend of mine experiences homicidal thoughts. She doesn't believe she would act upon it, but it definitely happens more than the average person (which, most people at least in one point in their lives experience homicidal thoughts). She is a diagnosed sociopath.
Mostly I feel she would need a hobby or a purpose in life.. Finding determination to achieve something is tough for some people already, in your friend's condition it's even more challenging
Samrat
Friends and family are important support system.. She also has to reach out for help
Samrat
how can you calm yourself when in distress.. i cant simply calm my nerves with breathing.. negative result
Depending on the severity of said distress it depends on how extensive your feelings of anxiety & panic are. So the most universally applicable "safe method", would be to attempt to change subjects, as in doing something possibly unrelated to keep busy mentally
Miguel
If that's not sufficient enough aside from the plethora of other alternatives I'm sure you've probably searched online. I'd like to understand a little better how "difficult" things become? Hyperventilating? Feeling like a heartattack? Or just overwhelming apprehension, worry, etc?
Miguel
what kind of distress? And how do you get a negative result from breathing?
iWasHacked
i feel worry, hyperventilating.. i cant control my breathing sometimes.. my bad past shows up, then i stupidly doubting myself a lot.. cant even look ppl in the eye n lock myself in my room. i try the breathing methods on google but it doesn't work, maybe i need someone to help me. such a pussy
Nicki
how the hell did this conversation get on my messages
mauReeN
meesages
mauReeN
idk
Nicki
I'm still trying to figure out what this is exactly.
Jonah
I have problems with hyperventilating as well so I can understand, so usually when I'm having an episode like that I concentrate intently on one object (ex. the floor, its design, the details in it, how it feels under my hands, etc).
iWasHacked
The Last time I had an episode it got intense enough where my extremites went numb and tingly and I couldnt see straight(probably close to fainting) with an intense pressure in my head and I had focused on the way the floor felt under my hands(stimulation helps).
iWasHacked
even though I couldnt really feel much but I had calmed down. Another thing that could possibly help is reciting over and over in your head if you are able to "I'm ok, just calm down,everything is fine you are going to be ok, you are safe breathe in and then out" this is usally what i do on the mild
iWasHacked
cases when I am just starting but when the hyperventilating gets intense freaking out from the loss of feeling is i think normal but its important to try and stay calm and try some stimulation for distraction. Other times my mother has helped to calm me down a lot, she usually was the one to help me
iWasHacked
so maybe ask for somebody to talk to you in that situation to distract your mind. Another thing I find is its best to sit down on your knees on the floor because if you faint you dont want to hit your head and usually when hyperventilating I surely cannot stand up straight so its just easier
iWasHacked
and you'll have one less thing to worry about in that situation. I really hope one of these can help you, its a very stressful situation and I really hope you can find some peace. Just remember that because something works for somebody else does not mean it will work for you so don't get upset if
iWasHacked
these dont help you just keep looking for something that works for you. You could also try some meditation outside of your episodes and maybe incorporate the meditation when you hyperventilate? Good luck!💛
iWasHacked
sorry for so many comments it wouldnt let me put it all in one comment lmao. Hope it helps.
iWasHacked
12345
Graced
I feel anticipating the situation and knowing that it could've been worse, can calm you down.. you basically need to be okay with whatever life throws at you and think ahead
Samrat
how do you get someone to stop smoking marijuana
you dont ..They need to want it for them self no one can make anything they don't want..They need to learn to have self control to live with out it..
Barrera
anyone**
Barrera
I lost 32 lbs in 6 weeks because all I could do was dry heave. Don't judge someone when you know nothing about them or their struggles
Tara
first you need to pinpoint why they are using it. There may be a health motive behind it.. if so marijuana is the most natural form of any pain relief available. It has several health benefits depending on the user and purposes.
Even
I get someone from smoking marijuana when I look at the bad side of it
austin
you know guys marijuana is a med but all meds will be bad if youll use it exessively
facts
Anxiety is why I stopped
Gianni
First question is. Why are you wanting to get somebody to quit? It could be purely their choice even knowing the side affects and if that's so you cant really do anything. Last, how long has this person been smoking and why is he or she smoking?
iWasHacked
What are some usual cognitive behavioral therapy routines for people dealing with addiction? More specifically, a case of a 21 y.o man with a comorbidity of marijuana addiction, addiction to sexual fantasy & pleasure, and an addiction to food. If you guys could help, I would appreciate it :)
what does marijuana do in the body of a person?
austin
What's the best way to keep ur cool when in a relationship, when u have a short temper for people playing dumb, insulting your intelligence as if they really think u r believing the crap there telling u? I mean its soooo aggravating for someone like me not to have broke him in half already but I want to do all I can to help him cuz I love him and I'm learning all I can to help us both
Pepole need to want change on there own, by u keep helping you are becoming an enabler.. Live by example and others will fallow..
Barrera
i'd say the relationship isn't healthy. if he is aware that he is hurting you by insulting you, than he does it to assert power. I'd say a healthy relationship needs to have an equal say in everything. In relationships, you often have to rely on the other. if he isnt there, take it as a sign.
Eduardo
you sho right Eduardo, anger can be controlled, it takes patience, strong mind and instantaneous thinking to, but from experience i believe narcacissm could be a factor! they tend to deny the fact of recognizing their faults to avoid the shame label, and so on...
Jo
He hits me starts fights then wonders why I stay away from h im.Right now I have no where be to go.I starting to get physically ill from the stress.Plus he never takes me anywhere and I left her by myself all the time.I never see people unless it's his friends.I know it's not good for me.I am alon
Vicki
For me How do I cope find inner peace.I cry all the time.
Vicki
Hello everyone.
Jonah
To find inner peace seek God 100% believe and you will never be the same again like crying all the time because that's what the devil wants from you and don't stuck up like just cry, be depressed it's just an example move forward not reverse and think & do with all your might the things that could
mary
possibly go right if you just conquer it with a burning desire, just dont stay stuck up explore there's so much things that u need to unravel
mary
Hello too.
mary
Hi, Todd I just want to share this to you. You see love is a two-way you give and receive its never a one-way road if you get what I mean. I know its hard for you but your significant other should understand you, it's not always him/her that needs to lend a caring, support, understanding. Your not
mary
dumb neither your significant other so both of you should talk about it and tell him/her what you've been through in the relationship. Don't hate on me Todd I'm just giving my opinion.
mary
What do you think the kind of relationship you have? All they tell you is indeed true and has point but are you clear with your relationship?
Dark
All he/she wants is to feel some importance so give him the importance he is grieving for and always when your in a conversation talk about his point of view and encourage him to talk about himself more.
Abdirahman
When you disagree something with him don't oppose him just say you maybe correct but this what I thought and then say your opinion. Just remember that you cannot teach a man anything; you can only help him to find it within himself. Be wiser than other people if you can; but do not tell them so.
Abdirahman
If they truly love you they will never hurt you. when you love someone you show you love them, not with words but with actions. show you they care, that you are important to them. From experience I have learnt that words mean nothing. you love someone, prove it
Jessamine
Just A fact me and Dark the way we said it or type in this chat Todd is not rude or try to be offensive. Just to clear things out.
mary
@Jo exactly! it's almost textbook. The longer he continues to get his ego fed, he'll continue to disrespect her. either he needs a therapist or someone to teach him that he isn't that important.
Eduardo
You know guys, LOVE is a very big thing to deal with. It's a thing that confuses everybody, a great topic for argument. I think the best way to deal with your problem is to know yourself.We can't keep on advising you like you should do this and that or believe this or that or follow this or that.
Dark
We perceive things differently, it's all clear. I'm not offending anyone in here. I'm sorry for my words. That's just how things go with my head. I'm not even good in love.
Dark
I agree. Short tempered people are evey where. I'm Short-tempered. People change when they want to. So, it could me a minute - days to years and they'll either change, or not. but the conclusion comes down to you both. Not how anyone ones judging the relationship.
Gianni
I can't see the conversations
i think its delete randomly
Uzair
yes probably
nazranna
hi
Fyke
hey
nazranna
ahoy wah gwaan
Tumo
Helloo everybody!
Jo
Hey☺
vidisha
Helo
Anna
hello
aravinth
Jo
I cant see the conversations
?
Sma
!
Uzair
put some glasses
?
Sma
Nicki
Guys can you give any idea what are the current challenges in educational psychology
Siena
how do you use reverse psychology?
dont use it
Eduardo
when dealing with kids,or young adolescent. Attempting to maybe alter the way they either percieve or feel about themselves, situation,person, ...
Jo
It is not the best method to use expecially in certain situations. It doesn't work on everyone and I know some cases where it has led to rifts.
im in accord,i guess it would depend on the situation and the way it's being expressed...opinion?
Jo
I 100% agree. I know cases where it has worked and cases where it has not. You need to analyse the situation before trying it out but once again it works on some people and not on others.
King
Hello, can I ask what are the current issues in educational psychology?
Siena
Oh sorry
Siena
Hi
Uzair
can anyone give me a personality test
look4 one online
Barrera
Nicki
***personalityperfect.com/test/free-personality-test/
Vincent
what is crisis intervention
Crisis intervention means immediate methods use to help an individual in a crisis situation.
vidisha
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