2OLE Variation of Parameters

Substituting this into the homogeneous equation and equating coefficients of like powers of t gives you an indicial equation in terms of the parameter r. Solving this equation will give you two values for r. These values will then help you find the two linearly independent solutions to the homogeneous equation, which you can then use with the variation of parameters method to solve the original non-homogeneous equation.In summary, to solve the given equation, you will need to use variation of parameters and find the two solutions to the homogeneous equation by solving the indicial equation.
  • #1
[V]
28
0
[tex]ty''-(t+1)y'+y=t^2[/tex]

I know I have to use variation of parameters to solve this.
But I am stuck and cannot figure out how to get the homologous equation!
[tex]y''-(1+\frac{1}{t})y'+\frac{1}{t}*y=t[/tex]

I don't know how to solve this homologous equation in this format.

Is it R^2+(1+1/t)R+1/t = 0 ?
How would I get my two solutions from this?

Thanks!
 
Last edited:
Physics news on Phys.org
  • #2
Whats wrong with the latex??
 
  • #3
[V];3246347 said:
[tex]ty''-(t+1)y'+y=t^2[/tex]

I know I have to use variation of parameters to solve this.
But I am stuck and cannot figure out how to get the homologous equation!
[tex]y''-(1+\frac{1}{t})y'+\frac{1}{t}*y=t[/tex]

I don't know how to solve this homologous equation in this format.

Is it R^2+(1+1/t)R+1/t = 0 ?
How would I get my two solutions from this?

Thanks!

Your homogeneous equation is

[tex]ty''-(t+1)y'+y=0[/tex]

This equation has a regular singular point at t = 0 suggesting you look for series solutions of the form

[tex]y = \sum_{n=0}^{\infty}a_nt^{n+r}[/tex]
 

1. What is the "2OLE Variation of Parameters" method?

The "2OLE Variation of Parameters" method is a mathematical technique used to solve differential equations involving one independent variable. It is an extension of the traditional Variation of Parameters method, and it allows for the solution of more complex and non-homogeneous equations.

2. How does the "2OLE Variation of Parameters" method differ from the traditional Variation of Parameters?

The "2OLE Variation of Parameters" method differs from the traditional method in that it involves two linearly independent solutions of the homogeneous equation, rather than just one. This allows for a more accurate and general solution to be obtained.

3. What types of differential equations can be solved using the "2OLE Variation of Parameters" method?

The "2OLE Variation of Parameters" method can be used to solve any type of first or second order differential equation with a single independent variable. This includes both homogeneous and non-homogeneous equations.

4. What are the steps involved in using the "2OLE Variation of Parameters" method?

The steps involved in using the "2OLE Variation of Parameters" method are as follows: 1. Find the general solution to the homogeneous equation using any standard method. 2. Find two linearly independent solutions to the homogeneous equation. 3. Use these solutions to find the particular solution to the non-homogeneous equation. 4. Combine the general solution with the particular solution to get the final solution to the equation.

5. What are the advantages of using the "2OLE Variation of Parameters" method?

The "2OLE Variation of Parameters" method has several advantages over other traditional methods of solving differential equations, such as: - It allows for a more general and accurate solution to be obtained. - It can be used to solve a wider range of differential equations. - It is relatively easy to understand and implement. - It does not require any special assumptions or conditions to be met.

Similar threads

A Solve PDE w/ Comsol 5.3: Numerical Solution & Time Evolution
    • Differential Equations
Replies
5
Views
1K
I Solution of delayed forcing function
    • Differential Equations
Replies
5
Views
1K
I Determining equilibrium solutions to differential equations
    • Differential Equations
Replies
16
Views
894
I Verifying properties of Green's function
    • Differential Equations
Replies
1
Views
752
I On error estimates of approximate solutions
    • Differential Equations
Replies
1
Views
666
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
1K
I Solving 2D Heat Equation w/ FEM & Galerkin Method
    • Differential Equations
Replies
7
Views
2K
MHB What is the integrating factor for solving this IVP?
    • Differential Equations
Replies
5
Views
1K
I On sub and super solutions: Teschl and others
    • Differential Equations
Replies
5
Views
653
MHB Second-Order Nonlinear Differential Equation
    • Differential Equations
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top