Help with finding particular solution of a 2nd order linear ode

• iqjump123
In summary: So we have:Y_{p}(x) = -e^{x} \int \frac{x e^{x}}{(e^{x}+1)^2}dx + e^{x} \int \frac{1}{e^{x}(1+e^{-x})}-\frac{1}{e^{x}(1+e^{-x})^2}dxUsing the substitution u = e^{-x}
iqjump123

Homework Statement

obtain the general solution y(x) of
y''-2y'+y=e^(2x)/(e^x+1)^2

Homework Equations

variation of parameters

The Attempt at a Solution

I have obtained the continuous equation.
I tried two methods of variation of parameters, but both of them got me stuck.

1. using the wronskian method. However, this method gave me an answer of 0.
2. Using the method involving setting a'(x) and b'(x) to 0- however, this method doesn't work if the complimentary solution involves a multiple of x (b'(x)xe^x).

This one seems to be more complicated than a simple constant of coefficients- wolfram got me a straightforward answer, however. Any help will be appreciated. TIA!

iqjump123 said:

Homework Statement

obtain the general solution y(x) of
y''-2y'+y=e^(2x)/(e^x+1)^2

Homework Equations

variation of parameters

The Attempt at a Solution

I have obtained the continuous equation.
I tried two methods of variation of parameters, but both of them got me stuck.

1. using the wronskian method. However, this method gave me an answer of 0.
2. Using the method involving setting a'(x) and b'(x) to 0- however, this method doesn't work if the complimentary solution involves a multiple of x (b'(x)xe^x).

This one seems to be more complicated than a simple constant of coefficients- wolfram got me a straightforward answer, however. Any help will be appreciated. TIA!

My personal preference in such cases is to always use the Green's Function method. In your case, let Ly(x) be the operator on the left of your DE. To solve Ly(x) = f(x), try solving LG(x) = delta(x-t) for G = G(x;t), where delta(x-t) = Dirac delta and t is just a parameter. Then Y(x) = integral_{t=-infinity..infinity} G(x;t) f(t) dt is a particular solution of LY(x) = f(x).

For more on Green's functions and the step-by-step solution of a problem almost like yours, see the pedagogical article http://www.mathphysics.com/pde/green/g15.html . This article also relates the Green's function to the Wronskian.

RGV

Hey Ray Vickson,

I appreciate your help. It is just that I haven't tried using greens function to solve an ODE, and therefore would like to see if there is an easier and/or more straightforward way?

The Wronskian method didn't give me zero. You should find that:

$$W(e^{x}, x e^{x}) = e^{2x}$$

Then the integrals are:

$$Y_{p}(x) = -e^{x} \int \frac{x e^{x}}{(e^{x}+1)^2}dx + xe^{x} \int \frac{e^{x}}{(e^{x}+1)^2}dx$$

1. How do I determine the particular solution of a 2nd order linear ODE?

The particular solution of a 2nd order linear ODE can be determined by using the Method of Undetermined Coefficients or the Method of Variation of Parameters. These methods involve finding coefficients and functions that satisfy the differential equation and any initial conditions given.

2. What is the difference between the Method of Undetermined Coefficients and the Method of Variation of Parameters?

The Method of Undetermined Coefficients involves assuming a particular form for the solution and solving for the coefficients using algebraic methods. The Method of Variation of Parameters involves using a particular solution found from the Method of Undetermined Coefficients and finding a complementary solution to satisfy the initial conditions.

3. Can I use the Method of Undetermined Coefficients for any 2nd order linear ODE?

No, the Method of Undetermined Coefficients can only be used for non-homogenous equations with constant coefficients. It cannot be used for non-constant coefficients or non-linear equations.

4. What is the particular solution for a homogenous 2nd order linear ODE?

For a homogenous 2nd order linear ODE, the particular solution is simply the zero function, as the equation only has the complimentary solution.

5. Are there any other methods for finding the particular solution of a 2nd order linear ODE?

Yes, there are other methods such as the Laplace transform method and the Power Series method. However, these methods are more advanced and are typically used for more complex equations.

• Calculus and Beyond Homework Help
Replies
2
Views
459
• Calculus and Beyond Homework Help
Replies
7
Views
712
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
3
Views
925
• Calculus and Beyond Homework Help
Replies
2
Views
406
• Calculus and Beyond Homework Help
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
3K