Solution of Euler Differential Equation Using Ansatz Method

In summary, the problem involves solving an Euler differential equation by using the ansatz y(x)=cx^{m}, where c and m are constants. After finding the values of m, it is determined that c cannot be found without initial conditions.
  • #1
dave4000
16
0

Homework Statement


Solve the euler differential equation

[tex]\x^{2}y^{''}+3xy'-3y=0[/tex]
[tex]
\int_X f = \lim\int_X f_n < \infty
[/tex]
by making the ansatz [tex]y(x)=cx^{m}[tex], where c and m are constants.

The Attempt at a Solution



[tex]y(x)0=c^{m}[tex]
[tex]y^{'}(x)=cm^{m-1}[tex]
[tex]y^{''}(x)=cm(m-1)^{m-2}[tex]

[tex]m(m-1)+3m-3=0[tex]
[tex]m^2+2m-3=0[tex]
[tex](m-1)(m+3)=0[tex]
[tex]m=-3 or m=1[tex]

Is this the solution or can c be found?
 
Last edited:
Physics news on Phys.org
  • #2
Latex isn't working on this post so here it is without Latex:

Homework Statement


Solve the euler differential equation

x^{2}y''+3xy'-3y=0

by making the ansatz y(x)=cx^{m}, where c and m are constants.

The Attempt at a Solution



y(x)=cx^{m}
y'(x)=cmx^{m-1}
y''(x)=cm(m-1)x^{m-2}

m(m-1)+3m-3=0
m^2+2m-3=0
(m-1)(m+3)=0
m=3 or m=-1

Is this the solution or can c be found?
 
Last edited:
  • #3
Latex maybe out due to technical problems.

Is one sure of the function before the y' term - 3x^{2}?
 
  • #4
*Correction made* :)
 
  • #5
Astronuc said:
Latex maybe out due to technical problems.

Is one sure of the function before the y' term - 3x^{2}?

This was merely a typo, the orignal problem still remains...
 
  • #6
so...er...c?
 
  • #7
since no innitial conditions were given i shall take irt that c cannot be found.
 

1. What is the Euler differential equation?

The Euler differential equation is a mathematical expression that describes the relationship between a function and its derivatives. It is named after the famous mathematician Leonhard Euler.

2. What is the general form of the Euler differential equation?

The general form of the Euler differential equation is y' = f(x,y), where y' represents the first derivative of the function y with respect to x and f(x,y) is a given function of both x and y.

3. What is the significance of the Euler differential equation?

The Euler differential equation is a fundamental tool in solving a variety of real-world problems in physics, engineering, and mathematics. It is also used to model dynamic systems and predict their behavior over time.

4. How is the Euler differential equation solved?

The Euler differential equation can be solved analytically or numerically using various techniques such as separation of variables, substitution, or using Euler's method. The specific method used depends on the given initial conditions and the complexity of the equation.

5. What are some applications of the Euler differential equation?

The Euler differential equation has numerous applications in fields such as physics, engineering, economics, and biology. It is used to model the motion of objects, population growth, electrical circuits, chemical reactions, and many other natural phenomena.

Similar threads

  • Calculus and Beyond Homework Help
Replies
2
Views
668
  • Calculus and Beyond Homework Help
Replies
6
Views
793
  • Calculus and Beyond Homework Help
Replies
18
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
132
  • Calculus and Beyond Homework Help
Replies
1
Views
465
  • Calculus and Beyond Homework Help
Replies
5
Views
447
  • Calculus and Beyond Homework Help
Replies
2
Views
61
  • Calculus and Beyond Homework Help
Replies
5
Views
595
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
953
Back
Top