Serious problem about Differential equation

In summary, Hi! I have 3 really problem here that i beg anyone who knows more to help me: 1. How to solve the equation of the kind xy'-3y=3 using power series. 2. I don't know what is exactly an ordinary point. They say that an ordianry point is a point at which an ordinary differential equation is "analytic",but i don't understand that term "analytic". 3. Is it important to look for the radius of convergence first, when you want to solve an ODE using power series? I always consider around x=0.
  • #1
kthouz
193
0
Hi! I have 3 really problem here that i beg anyone who knows more to help me:
1. How to solve the equation of the kind xy'-3y=3 using power series.
I usually solve kind pf homogeneous equations. Here the problem rises when i want to equalise all the coefficient of x^n to 0. Because of that 3 there i don't now how to procede.
2. I don't know what is exactly an ordinary point. They say that an ordianry point is a point at which an ordinary differential equation is "analytic",but i don't understand that term "analytic".
3. Is it important to look for the radius of convergence first, when you want to solve an ODE using power series? I always consider around x=0.
 
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  • #2
kthouz said:
Hi! I have 3 really problem here that i beg anyone who knows more to help me:
1. How to solve the equation of the kind xy'-3y=3 using power series.
I usually solve kind pf homogeneous equations. Here the problem rises when i want to equalise all the coefficient of x^n to 0. Because of that 3 there i don't now how to procede.
That is exactly why memorizing formulas without recognizing them is a mistake!

If this were xy'- 3y= 0,you would know what to do? Then you would write [itex]y(x)= \sum_{n=0}^\infty a_n x^n[/itex] so that [itex]y'(x)= \sum_{n=1}^\infty n a_n x^{n-1}[/itex] and your equation would be
[itex]xy'- 3y= \sum_{n=1}^\infty n a_n x^n- \sum_{n=0}^\infty 3a_n x^n= 0[/itex]
Notice that the first sum is from 1 to infinity while the second sum is from 0 to infinity. That happens because when you differentiate a power series, you lose the first term. Of course, since you say you can solve homogenous equations, you know that you need to treat that case separately: If n= 0, 3a0= 0 so a0= 0. If n> 0 then
nan- 3an= (n- 3)an= 0 which makes the solution kind of trivial! Either an= 0 or n= ? (Of course, it's a separable equation and easy to solve that way.)

The only difference now is that you have
[itex]xy'- 3y= \sum_{n=1}^\infty n a_n x^n- \sum_{n=0}^\infty 3a_n x^n= 3[/itex]
You can think of that right hand side as 3x0 in order to match it to the left hand side. Again, do the n=0 case separately: 3a0= 3 so a0= 1 now. Do you see what to do if you had x2 or 3+ x2 on the right side? For some more complicated function, like ex or cos(x), expand them in their Taylor series and then equate corresponding coefficients.

2. I don't know what is exactly an ordinary point. They say that an ordianry point is a point at which an ordinary differential equation is "analytic",but i don't understand that term "analytic".
A real valued function is analytic at a point if its Taylor's series exists at that point (so it must be infinitely differentiable in order to have a Taylor's series) and converges to the function on some neighborhood of the point. Caution- there exist functions that "have" Taylor's series but the radius of convergence is 0. There exist functions that have Taylor's series that converge for all x but not to the function! Fortunately for you almost all, if not all, of the examples you will see are "not analytic" because they have a factor in the denominator that goes to 0 at that point.

3. Is it important to look for the radius of convergence first, when you want to solve an ODE using power series? I always consider around x=0.
I wouldn't consider it "necessary", although it might save you some work if you found the radius of convergence was 0!

You shouldn't consider only "around x= 0". You might well have a problem where the "initial condition" is given at some point other than x= 0. If you were told that y(a)= 1, satisfying that condition will be much easier if you wrote [itex]y= \Sum a_n (x- a)^n[/itex]. Or you could "shift" your variable: write t= x- a and rewrite the equation in terms of t rather than a.
 
  • #3
Thanks!
And can someone show me (give me hint on) how to find legendre polynomials on 1st and 2nd degrees using (1-2xh+h^2)^(-1/2)=∑h^n P_n(x) .
 

Related to Serious problem about Differential equation

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It involves variables, functions, and their derivatives, and is used to model and solve various real-world problems in science and engineering.

2. Why are differential equations important?

Differential equations are important because they provide a powerful tool for describing and predicting the behavior of dynamic systems in various fields such as physics, engineering, economics, and biology. They also play a crucial role in the development of new technologies and advancements in science.

3. What are the different types of differential equations?

Differential equations can be classified into several types, including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs). ODEs involve one independent variable, while PDEs involve multiple independent variables. SDEs take into account random fluctuations in a system.

4. How are differential equations solved?

Differential equations can be solved analytically or numerically. Analytical solutions involve finding an exact solution using mathematical techniques such as separation of variables, integration, and series expansion. Numerical methods involve using computers to approximate the solution through iterative calculations.

5. What are some applications of differential equations?

Differential equations have numerous applications in various fields, such as modeling population growth, predicting weather patterns, designing electrical circuits, analyzing fluid flow, and understanding chemical reactions. They are also used in the development of mathematical models for complex systems and in data analysis.

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