Euler equations, Mathematics

Assignment Help:

Euler Equations - Series Solutions to Differential Equations

In this section we require to look for solutions to,

ax2 y′′ + bxy′ + cy = 0

around x0  = 0. These kinds of differential equations are termed as Euler Equations.

Recall from the earlier section which a point is an ordinary point if the quotients the quotients,

(bx)/(ax2) = (b/ax)                               and c/(ax2)

Above equations have Taylor series around x0 = 0. Though, due to the x in the denominator neither of these will contain a Taylor series around x0 = 0 and so x0 = 0 is a particular point.  Therefore, the method from the earlier section won't work as it needed an ordinary point.

Though, this is possible to find solutions to this differential equation that aren't series solutions. Here we start off through assuming that x>0 (the purpose for it will be apparent after we work the first illustration) and which all solutions are of the form,

y (x) = xr    ............... (2)

 Here plug this in the differential equation to find,

ax2(r)(r-1) xr-2 + bx (r) xr-1 + cxr = 0

ar(r-1) xr + b(r) xr + cxr = 0

(ar(r-1) + b(r) + c)xr = 0

Now, we assumed that x>0 and so this will only be zero if,

ar ( r -1) + b (r) +  c = 0   ...................       (3)

Therefore, solutions will be of the form (2) given r is a solution to (3). That equation is a quadratic in r and thus we will have three cases to look at: Distinct Roots, Real and Double Roots and Quadratic Roots.


Related Discussions:- Euler equations

Times tables, how can you memorise you times facts

how can you memorise you times facts

Daily Math, Six times as many people voted in the 2012 election as in the 2...

Six times as many people voted in the 2012 election as in the 2008 election.If 162 people voted in 2008,how many people voted in both elections?

Find the ratio of line segment, Find the ratio in which the line segment jo...

Find the ratio in which the line segment joining A(6,5) and B(4,-3) is divided by the line y=2           (Ans:3:5) Ans :   Let the ratio be k:1 x = 4 k + 6/ k +1 y

Ravens played 25 home games how many games did they win, The Ravens played ...

The Ravens played 25 home games this year. They had 9 losses and 2 ties. How many games did they win? Eleven games are accounted for along with the losses and ties (9 + 2 = 11)

Comparing, compare 643,251 633,512 and 633.893 the answer is 633.512 what i...

compare 643,251 633,512 and 633.893 the answer is 633.512 what is the question

Number and operations, 1a.if the williams spend $385 a month on food what i...

1a.if the williams spend $385 a month on food what is their monthly income

Sum, As1212uestion #Minimum 100 words accepted#

As1212uestion #Minimum 100 words accepted#

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd