00:00so hello everyone we are going to start mathematical methods
00:10first we will start with the different terminology that we should know before starting this so
00:35starting with the first what is the ordinary differential equation OD so talking about the OD so
00:50in ordinary differential equation we deal with a function that has one variable and its derivative
00:59with respect to that variable so for example TY by DX is equal to minus KY so it has a one dependent
01:22variable and one independent variable and now talking about the partial differential equation
01:40so it has one dependent variable and two or more independent variable so for example DY by no let's say
01:57DU by DT is equal to K DTU by DX2 so in partial differential equation like
02:09there are two or more independent variable okay so now talking about the order of a differential equation
02:26so to find the order of differential equation the answer will be the highest derivative that appears
02:35so for example so for example plus 2Y double R plus Y is equal to 0 so the order of a differential equation
02:47equation here will be a 3 because the this has the highest power it is 2 here it is
03:050 so order of a differential equation here is 0 so order of the differential equation here is 3 now talking about the next terminology so now we have a initial value problem
03:16so let's say in initial value problem
03:28so let's say in initial value problem
03:34so let's say in initial value problem
03:46we have the equation we have the equation so we will be given with the details of a point so that we can get the exact equation and for boundary value problem we will be given the value of the two extreme values so that we can find the equation so let's say in initial value problem Y double dash plus Y is equal to 0 so we will be given the value of Y
03:53Y at 0 and also the value of Y at 0 is equal to 0 and also the value of Y at 0 is equal to 0 and putting these two
04:00after solving this two after solving this equation and putting these two after solving this equation and putting this value we will get the exact equation
04:07So, let's say in initial value problem, y double dash plus y is equal to 0.
04:13So, we will be given the value of y at 0 and also the value of y dash at 0 is equal to 0.
04:21And putting this two, after solving this equation and putting this value, we will get the exact
04:28equation.
04:29And for boundary value problem, let's say the equation is y double dash plus y is equal
04:35to 0.
04:35So, here we will have the value at both the extremes.
04:40So, let's say y 0 is equal to 0 and y of pi is equal to 0.
04:48So, this is the difference between the initial value problem and boundary value problem.
04:54Now, talking about the...
05:05Now, next, talking about the linear differential equation.
05:14So, linear differential equation, so, means the unknown function and its derivative are
05:22never multiply together or put inside non-linear functions like sine y, sine x, cos x.
05:33So, general linear problem is...
05:36So, linear...
05:37So, linear...
05:39Linear...
05:41So, linear...
05:55So, linear differential equation is much easier to solve.
06:09So, like 2 y raise power...
06:13No, let's say like this...
06:18An x y raise power n plus...
06:23An minus 1 x y raise power n minus 1 is equal to g of x.
06:31So, these are linearly dependent.
06:33And...
06:34Now, talking about the next term...
06:39Homogeneous versus non-homogeneous.
06:41So...
06:42So, like y double dash plus y is equal to 0.
06:56This is a homogeneous.
06:59And if y double dash plus y is equal to sine x.
07:03So, this will be a non-homogeneous.
07:06And now, talking about the constraint versus non-constraint coefficients.
07:21So, for constraint coefficient...
07:33Let's say...
07:34Constant versus non-constraint...
07:40So, for...
07:43So, we have to look for the equation.
07:45So, the example of the coefficient...
07:48A constant coefficient is...
07:52So, here we are talking about these coefficients.
08:03So, these are constants.
08:05And...
08:06Example of the non-constraint coefficient are like...
08:09X square y double dash plus y dash plus y is equal to 0.
08:14This is the example of the non-constraint coefficient.
08:17Because the value of the X can vary.
08:19Yes.
08:20So, now...
08:23Let's summarize.
08:25What is the difference between the ordinary versus partial differential equation?
08:29So, it depends on how many independent variables we have.
08:32If there is one independent variable, it will be ordinary.
08:35And if it have two or more independent variables...
08:39So, it will be partial differential equation.
08:41Order of differential equation means the highest power it have.
08:45Independent versus boundary value problem.
08:48Independent means the value will be given of a single point.
08:51And for boundary, the value will be given of the two extremes of points.
08:58Linear versus non-linear.
09:02So, for linear versus non-linear, like...
09:06For linear, what are the...
09:09Like, our power multiplication allowed?
09:13So, like, for linear, it is linearly dependent.
09:17And for non-linear, like...
09:20It must be, like, multiple of different powers.
09:24Homogeneous versus non-homogeneous.
09:26So, for homogeneous, we can see that...
09:30This is the example of homogeneous, and this is the example of non-homogeneous.
09:33And for it, constant is non-constant.
09:35Here, these are constant values, and these are the non-constant values.
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