solving second order ordinary differential equations.

As the demand for precision in scientific and non-scientific experiments increased, maths evolved accordingly.Intially there were algebraic equations and now there are differential equations

*see mathematical modelling for a clear picture want I mean.

Considering you have an insight of differentiation and integration a bit, I would like to move on with introducing differential equations.

Many times mathematical modelling ended up generating multi-variable (more than two) algebraic expressions, so these guys are called equations not just functions. Functions are supposed to have only two variables let it be either implicitly or explicitly the point is that they must give a peculiar output for every input.

Further we came across an algebraic expressions that has both derivative and the function of a particular function [rate suggesting dy/dx and relation suggesting y=f(x)].they were multivariable sometime and in fact many time, so the name differential equation not differential function.

It was initially a scientific demand to generate the relation between the variables which were part of the differential equation, considering the independent and dependent variables off course.

So learning to solve differential equations has got meaning : ) , any how to solve differential equation is to find the relation of variables involved, its generally a function.

They say there are 23 ways to solve differential equations, I don’t know exactly, we may figure it out some other day.

As algebraic expressions, differential equations are also classified

PIC

First order first degree equations can be solved using fallowing methods.

·         Variable separable method

·         Homogenous form

·         Exact form

·         Linear form( using integrating factor)

Second order first degree equations (both homogenous and in-homogenous) can be solved employing fallowing methods.

·         Method of undetermined co-efficient

·         Inverse differential operator

·         Variation of parameters

Basically one can catch up the idea of differential operator if he had a meaningfully understanding of what differentiation actually is? 

  With that idea in mind one can clearly and easily reduce any second order linear equation (even higher order) as such>

                                                           f(d)y =g(t)

                                                                

                                      And                     y=g(t)/f(d)

Hurray! sign of getting a relation, but the only problem is f(d). Sometimes you might consider it as an inverse differential operator and work off till you get what you needed, but it sucks a lot, you can’t integrate all the time.

Considering if  y  were to be of the form e^ax  (we decided so because of the hint given by g(t))  {{{[let it may be some constant multiplied or divided to e^ax  our assumption works, thanks to linearity of differential operator]}}}

We found out that f(d) was equal to f(a).

If ‘a’ made f(d) inhomogeneous it was a good move, if it made f(d) homogenous. It suggests that ‘a’ was one of the roots of f(d) and this one too sucks. . .

Hey, watch out 

                                If             f(a)=0 

                       Means                y=infinity,   

 Which is ridiculous to believe and in fact we dealt such situations previously, remember limits. . . . .when things turned out awkward we simplified them and got reasonable response

EXAMPLE:>

Similary  f(d) was split into (d-b)(d-c )  and at any time if a=b, that term along with the numerator was simplified and voila we got a reasonable response. BONUS  the remaining term at the denominator aka (d-c) was found to be equal to f”(a) .

EXAMPLE:>

Oh no, the relation we got from above method doesn’t satisfies the given differential equation ,so were actually we went wrong. . . .lets checkout together . . . . . . .

Ha ha here it is (there it was infact {G})    unknowingly we are considering the roots of the homogenous f(d) i.e  c and b

One more time , . . clearly

m^2+2m+5     if m=4 [a=4]       ===============  29

m^2+3m+1       “     “     “           ===============  29             ? m=D=operator

m^2+5m-7       “     “     “            ===============  29

Plugging the same value of ‘a’ into so many second order linear equations ended up giving  the same output .

In fact the only loop hole in our solution was that we were unable to attain the peculiar f(d) we were supposed to get .

Every equation is peculiar because of its roots.

In fact they have a brilliant role to play while making the equation go inhomogeneous deciding the exact value of inhomogenity (RHS)

So finally it’s obvious to consider the homogenous solution of f(d).

Thus the complete solution of the given differential equation is the sum of this homogenous f(d)’s solution (usually regarded as general solution) and  y which we got initially(usually regarded as particular solution)

VARIATION OF PARAMETER

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