Linear second order non-homogeneous ODE question

In summary, the solution to the ODE is y = yh + yp. The homogeneous solution is y = c1 + c2e-2x. For the particular solution, I have been using the method of undetermined coefficients. c3e-2x won't work as it is not linearly independent of the homogeneous solution. So I guess c3xe-2x.
  • #1
Malby
16
0
Determine the general solution to the ODE:

y'' + 2y' = 1 + xe-2x

I know the solution will be of the form y = yh + yp. The homogeneous solution is y = c1 + c2e-2x.

For the particular solution, I have been using the method of undetermined coefficients. c3e-2x won't work as it is not linearly independent of the homogeneous solution. So I guess c3xe-2x.

So y' = c3e-2x(c3 - 2c3x)

And y'' = 4c3e-2x(x - 1)

Following on from this I end up with a particular solution:

yp = ((1 + xe-2x)/(-2e-2x))xe-2x

However wolfram disagrees with me. My question is, is the method of undetermined coefficients ok to use for this ODE? Or should I be using the method with the Wronskian (it's name escapes me at the moment)

Cheers
 
Physics news on Phys.org
  • #2
If you put y(x) = C1(x) + C2(x)*exp(-2x) in the DE, you eventually arrive at a first-order DE in v = dC1/dx + exp(-2x)*dC2/dx.

RGV
 
  • #3
Ray Vickson said:
If you put y(x) = C1(x) + C2(x)*exp(-2x) in the DE, you eventually arrive at a first-order DE in v = dC1/dx + exp(-2x)*dC2/dx.

RGV

I'm not sure I understand this. Are you able to explain it a little more?

Cheers
 
  • #4
I'd have to dig out my book or notes to make sure, but I think your particular solution will also need an e-2x term, so it should be of the form Ae-2x + Bxe-2x. I wouldn't use c's for yh since they're already being used as the constants of integration for yh.
 
  • #5
For the method of variation of the parameters, read http://en.wikipedia.org/wiki/Variation_of_parameters.

This problem can be solved for y' as the y term is missing. Denote z=y' and solve the first-order equation z'+2z=1+xe-2x, then integrate z=y' to get y.

ehild
 
  • #6
ehild said:
For the method of variation of the parameters, read http://en.wikipedia.org/wiki/Variation_of_parameters.

This problem can be solved for y' as the y term is missing. Denote z=y' and solve the first-order equation z'+2z=1+xe-2x, then integrate z=y' to get y.

ehild

Aha! Of course. It's all so simple once you know what to do... :rofl:

Thanks very much!
 

1. What is a second order linear non-homogeneous ODE?

A second order linear non-homogeneous ODE (ordinary differential equation) is an equation that involves a second derivative of a function, as well as the function itself, and also includes a non-zero function on the right-hand side. It is written in the form: y''(x) + p(x)y'(x) + q(x)y(x) = g(x), where p(x) and q(x) are functions of x and g(x) is a non-zero function of x.

2. How is a linear second order non-homogeneous ODE solved?

There are several methods for solving a linear second order non-homogeneous ODE, including the method of undetermined coefficients, variation of parameters, and Laplace transforms. These methods involve finding a particular solution to the non-homogeneous equation and then combining it with the general solution to the corresponding homogeneous equation.

3. What is the difference between homogeneous and non-homogeneous ODEs?

A homogeneous ODE is one in which all terms contain the dependent variable and its derivatives, while a non-homogeneous ODE contains additional terms that do not involve the dependent variable or its derivatives. In other words, the right-hand side of a homogeneous ODE is equal to zero, while the right-hand side of a non-homogeneous ODE is not equal to zero.

4. Can a linear second order non-homogeneous ODE have multiple solutions?

Yes, a linear second order non-homogeneous ODE can have multiple solutions. This is because the general solution to a non-homogeneous ODE is a combination of the particular solution and the general solution to the corresponding homogeneous equation, and there can be multiple particular solutions that satisfy the non-homogeneous equation.

5. What are some real-life applications of linear second order non-homogeneous ODEs?

Linear second order non-homogeneous ODEs have many applications in science and engineering, including in the fields of physics, chemistry, and biology. They are used to model various processes such as oscillations, electrical circuits, and chemical reactions. They are also commonly used in signal processing and control systems.

Similar threads

Variation of parameter VS Undetermined Coefficients
    • Calculus and Beyond Homework Help
Replies
7
Views
500
Show that ODE is homogeneous, but I don't think it is
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Method of undetermined coefficients -- Help please
    • Calculus and Beyond Homework Help
Replies
3
Views
992
Avoid unpleasant integrals in solving IVP
    • Calculus and Beyond Homework Help
Replies
3
Views
570
Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##
    • Calculus and Beyond Homework Help
Replies
7
Views
706
Understanding Wronskian of solutions to homoeneous 2nd order linear DE
    • Calculus and Beyond Homework Help
Replies
2
Views
277
Question ODE non-homogeneous Linear
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Finding the homogenous solution of Var.Coeff. 2nd Order ODE
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
Second order non homogeneous ODE, IVP
    • Calculus and Beyond Homework Help
Replies
1
Views
2K
Y′′=−20⋅4x^3, Second order linear ordinary DE
    • Calculus and Beyond Homework Help
Replies
9
Views
1K

Hot Threads

Recent Insights

Back
Top