Euler equations, Mathematics

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.

Posted Date: 4/11/2013 2:25:47 AM | Location : United States







Related Discussions:- Euler equations, Assignment Help, Ask Question on Euler equations, Get Answer, Expert's Help, Euler equations Discussions

Write discussion on Euler equations
Your posts are moderated
Related Questions
Write down the system of differential equations for the population of both predators and prey by using the assumptions above. Solution We will start off through letting that

Application Practice Answer the following questions. Use Equation Editor to write mathematical expressions and equations. First, save this file to your hard drive by selecting Sav

Q. How to calculate Probability of event? Ans. What chance do I have to toss the coin and get a head? You might think 50-50, 50%. What about tossing it 5 times and getting

Describe Segments, Rays, Angles, and Triangles We now define some more basic geometric figures. 1. Segments Definition A segment is the set of two given points and all the


Another aid that can help children practise subtraction is the number strip. TGS can be used to improve their ability to count backwards. For example, subtracting 4 from 9 means


Any point on parabola, (k 2 ,k) Perpendicular distance formula: D=(k-k 2 -1)/2 1/2 Differentiating and putting =0 1-2k=0 k=1/2 Therefore the point is (1/4, 1/2) D=3/(32 1/2

There are 81 women teachers at Russell High. If 45% of the teachers in the school are women, how many teachers are there at Russell High? Use the proportion part/whole = %/100.

how smart do u have to be to get into google