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Thank you for visiting our site! You landed on this page because you entered a search term similar to this: Solving Non linear differential equations. We have an extensive database of resources on Solving Non linear differential equations. Below is one of them. If you need further help, please take a look at our software "Algebrator", a software program that can solve any algebra problem you enter!
Differential EquationsA differential equation is an equation which contains the derivatives of a variable, such as the equation
Here x is the variable and the derivatives are with respect to a second variable t. The letters a, b, c and d are taken to be constants here. This equation would be described as a second order, linear differential equation with constant coefficients. It is second order because of the highest order derivative present, linear because none of the derivatives are raised to a power, and the multipliers of the derivatives are constant. If x were the position of an object and t the time, then the first derivative is the velocity, the second the acceleration, and this would be an equation describing the motion of the object. As shown, this is also said to be a non-homogeneous equation, and in solving physical problems, one must also consider the
First Order Homogeneous DEA first order homogeneous differential equation involves only the first derivative of a function and the function itself, with constants only as multipliers. The equation is of the form
and can be solved by the substitution
The solution which fits a specific physical situation is obtained by substituting the solution into the equation and evaluating the various constants by forcing the solution to fit the physical boundary conditions of the problem at hand. Substituting gives
General Solution to a D.E.The general solution to a differential equation must satisfy both the homogeneous and non-homogeneous equations. It is the nature of the homogeneous solution that the equation gives a zero value. If you find a particular solution to the non-homogeneous equation, you can add the homogeneous solution to that solution and it will still be a solution since its net result will be to add zero. This does not mean that the homogeneous solution adds no meaning to the picture; the homogeneous part of the solution for a physical situation helps in the understanding of the physical system. A solution can be formed as the sum of the homogeneous and non-homogeneous solutions, and it will have a number of arbitrary (undetermined) constants. Such a solution is called the general solution to the differential equation. For application to a physical problem, the constants must be determined by forcing the solution to fit physical boundary conditions. Once a general solution is formed and then forced to fit the physical boundary conditions, one can be confident that it is the unique solution to the problem, as gauranteed by the
Boundary ConditionsThe boundary conditions on a differential equation are the constraining values of the function at some particular value of the independent variable. For example, if the equation involves the velocity, the boundary condition might be the initial velocity, the velocity at time t=0. In order to have a complete solution, there must be a boundary condition for each order of the equation - two boundary conditions for a second order equation, only one necessary for a first order differential equation. If a solution to a differential equation is found which satisfies all the boundary conditions, then it is the only solution to that equation - this is called the uniqueness theorem. Therefore, a reasonable approach to finding solutions to differential equations in physical problems is to use a trial solution and try to force it to fit the boundary conditions. If successful, then this approach finds the unique solution.
For the differential equations applicable to physical problems, it is often possible to start with a general form and force that form to fit the physical boundary conditions of the problem. This kind of approach is made possible by the fact that there is one and only one solution to the differential equation, i.e., the solution is unique.Stated in terms of a first order differential equation, if the problem
meets the condition such that f(x,y) and the derivative of y is continuous in a given rectangle of (x,y) values, then there is one and only one solution to the equation which will meet the boundary conditions.
Differential Equation Terminology
Some general terms used in the discussion of differential equations:
Order: The order of a differential equation is the highest power of derivative which occurs in the equation, e.g., Newton's second law produces a 2nd order differential equation because the acceleration is the second derivative of the position.
Linear and nonlinear: A differential equation is said to be linear if each term in the equation has only one order of derivative, e.g., no term has both y and the derivative of y with respect to time. Also, no derivative is raised to a power.
Homogeneous and nonhomogeneous: A differential equation is said to be homogeneous if there is no isolated constant term in the equation, e.g., each term in a differential equation for y has y or some derivative of y in each term.


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Recently published in Microbiology, the study from scientists at Teagasc (the Irish Agriculture and Food Development Authority), University College Cork and the Alimentary Pharmabiotic Centre examined whether a lactobacillus strain with CLA (conjugated linoleic acid) influenced fat tissue composition in mice.
Previous research from Racine et al. (2010) and Thom et al. (2001) has linked fatty acid t10, c12 CLA consumption with decreased body fat in humans, while other studies suggest that this type of fatty acid inhibits colon cancer cell growth.
Probiotics assists CLA metabolism
In this study, scientists transferred an enzyme-encoding gene from skin bacterium P.acnes to Lactobacillus paracasei and induced it to produce CLA t10, c12, which fed to mice resulted in a fourfold increase of CLA in mouse fat tissue composition (against a second probiotic control strain) showing that live bacteria intake affects metabolism at remote body sites.
The researchers wrote: “t10, c12 CLA has been shown to be beneficial to host health…and has been demonstrated to alter body composition by reducing the fat content and increasing the lean body mass in animal studies, and also in some human studies.”
Since one well-known side-effect of higher doses of CLA is liver steatosis (abnormal fatty lipid retention in human cells), they added: “The lower dose of CLA produced by a probiotic strain may be a reliable solution to control these adverse dietary effects.”
Fat is not inert
Study leader Dr. Catherine Stanton said: “CLA has already been shown to alleviate non-alcoholic fatty liver disease that often accompanies obesity. Therefore, increasing levels of CLA in the liver by ingestion of a probiotic strain is of therapeutic relevance.”
“Furthermore, fat is not an inert layer around our bodies, it is active and pro-inflammatory and is a risk factor for many diseases, including cancers. The work shows that there is potential to influence this through diet-microbe-host interactions in the gut.”
Two groups of eight-week old mice were fed CLA-supplemented diets (2% of weight) for 8 weeks: in combination with either recombinant t10, c12 CLA-producing Lactobacillus paracasei (8 mice) or a non-producing CLA control strain of the probiotic (8 mice).
A third group (5 mice) was fed CLA alone, while tissue fatty acid composition was assessed by gas liquid chromatography at the end of the trial, after the mice had been humanely killed and their organs removed.
Source: Microbiology, (published online ahead of print, December 22 2010) DOI: 10.1099/mic.0.043406-0
'Recombinant lactobacilli expressing linoleic acid isomerase can modulate the fatty acid composition of host adipose tissue in mice’
Authors: Cody, R-C; Stanton, C.; O’Mahony, L.; Wall, R. ; Shanahan, F. ; Quigley, E.M; Fitzgerald, G.F; Paul Ross, R.


Bedbugs are making a comeback and the bedtime rhyme to sleep tight and not let the bedbugs bite is no longer quite so charming.
Bedbugs are a small, reddish-brown insect that feed on the blood of sleeping people or animals. At about the size of an apple seed, these tiny creatures prefer dark places during the day and often hide between the mattress and box spring, behind the headboard and even inside wall cracks or crevices. The Centers for Disease Control and Prevention (CDC) reports that although bedbugs may travel up to 100 feet for a meal, they typically live within eight feet of where people sleep.
Close to Home
According to the National Pest Management Association, bedbugs are most commonly found in single-family homes, followed by apartments or condominiums and hotels. However, any place where people sleep would be attractive to bedbugs. Places with many travelers make it more likely that bedbugs can stow away in luggage or folded clothing and infest a new location.
Itchy bumps can take up to 14 days to appear after the bite occurs. There are often several bites together, and may be clustered or in a line. Dr. Bosch recommends cleaning bites with soap and water, and avoiding scratching. Over-the-counter anti-itch cream can help if the urge to scratch is strong.
“If the itch becomes unbearable or infected, it might be time to see your doctor,” said Bosch.
Bedbug bites are easy to confuse with flea bites. Bedbug bites are usually redder. Check thoroughly around sleeping areas if you suspect a pest infestation. Bedbugs reproduce quickly so do not delay looking for the source of any bite marks.
Signs of Bedbugs
You can check for bedbugs in between mattresses and under fitted sheets or bed skirts. Use a flashlight to check behind the headboard and in crevices along baseboards. You can also check the seams of upholstered furniture near the sleeping area.
Evidence of bedbugs include rusty brown spots of feces and molted exoskeletons left in dark hiding spaces. You may also see blood spots on your sheets if you’ve been bitten while you sleep.
Prevention on the Road
If you are checking into a hotel or any shared sleeping space, leave your luggage in the car or place it in the bathroom while you check the room for signs of bedbugs. If you are worried there might be bedbugs on your luggage when you return home, you can treat your luggage before you take it in the house. The Environmental Protection Agency (EPA) reports that placing items in a freezer set to 0° F for four days will kill bedbugs, as will steam temperatures of 130° F.
Items that you buy used or at garage sales should also be inspected for bedbugs. Dr. Bosch recommends washing things that are machine washable as soon as possible, and drying them on the hottest setting for at least 20 minutes.
Treating an Infestation
If you find you have bedbugs, you’ll need a comprehensive strategy to get rid of them. According to the EPA, bedbugs have become resistant to some pesticides and can hide where chemicals don’t reach them, so a one-time treatment like a fogger won’t eliminate the problem. A professional exterminator can manage the process for you, or the EPA has planning tools to help if you want to do it yourself.
“Bedbug bites aren’t dangerous but the bugs should be taken care of as soon as possible to avoid an infestation,” said Bosch.
Family medicine physician Stephanie Bosch, DO, sees patients of all ages at the Samaritan Waldport Clinic in Waldport, Oregon.
Learn more about bedbugs in our health library.
Preparing for a trip? Visit the International Travel Clinic and learn other ways to stay healthy at your destination.
If you need medical attention right away, visit samhealth.org/carenow to find out your nearby options for urgent care, walk-in and e-visits.


If you have been in a diagnostic testing facility, a pediatrician’s office, or a fertility clinic recently, then you might have noticed signs promoting cord blood banking. The medical community is really promoting it as much as it can.
Wait a minute. Just what is cord blood banking anyway?
After the mother gives birth, doctors extract hematopoietic stem cells, or HSCs, from the newborn umbilical cord, which survive outside the body and can be preserved for decades. Once the cord blood is banked, either in a private or public banking system, it can later be used to help cure a myriad of different diseases and illnesses, thanks to the fresh red and white blood cells.
Sounds like an incredible medical breakthrough, correct? Yes, it does. However, there are plenty of misconceptions about cord blood that prompt parents to reject the procedure.
Here are four misconceptions about cord blood that parents-to-be need to know:
1. No, It Doesn’t Hurt the Baby
One of the most common myths about cord blood banking is that parents think it hurts the baby. But the truth is this: it doesn’t the baby in the slightest.
In fact, the entire procedure, which only takes a couple of minutes, is painless and unobtrusive. It will never interfere with your initial bonding moments. It is simple and risk-free.
There is no reason to fear that your baby boy or girl will be hurt.
2. No, It Won’t Cure All Sicknesses
Yes, the medical industry has high hopes for cord blood. The results are already promising, and there are multiple case studies proving its efficacy.
That said, despite the optimism of doctors, nurses, and the entire medical community, it isn’t a cure-all method. At least, not right now.
Sure, it won’t cure cancer today, but with more research it will cure a wide array of diseases tomorrow. With the research done in the last few years, cord blood has proven better than bone marrow, they are more flexible than adult stem cells, they can be used in transplants, and they are natural treatments.
3. No, Not All Cord Blood Banks Are the Same
Another common misconception is that cord blood banks are all created equal. This is untrue.
There are generally two types of cord blood banks: private and public.
Public cord banks are for public use that’s free of charge and can be used by anyone else (there are some exceptions at certain hospitals in Canada). Private cord banks store your baby’s blood for future use, but it comes with an expensive monthly fee.
4. No, Your Baby Can Use His or Her Cord Blood
It is true that other sick patients can use your baby’s cord blood, which has some parents concluding that only other people can take advantage of their son or daughter’s cord blood stem cells. This is incorrect.
A large number of stem cell transplants using a person’s own cells are completed every single year for diseases like lymphoma or tumors.
Yes, if you donate the cord blood, you may lose access to it. But it doesn’t mean that your child’s cord blood can’t be used for a health scare of their own.
Cord blood is only advancing with each passing day, week, month, and year. The researchers examining its efficacy and potential uses are offering incredible results with each new study or test. But this can only be achieved through the heroic consent of fathers and mothers who are allowing doctors to extract stem cells from their child’s umbilical cord blood.
Like anything else relating to their children, parents may be apprehensive, worried, and anxious about what could possibly happen. Thanks to guidance, research, and conversations, moms and dads took the right step and permitted cord blood donations to take place.