Second-order nonhomogeneous diff-eq

In summary, the conversation discusses solving second-order differential equations with a non-homogeneous term, specifically the equation u'' + a^2*u = cos(bx). One participant mentions using the method of undetermined coefficients to find a particular solution, while another suggests using euler's formula to split the real and imaginary parts of the equation. The conversation ends with a preference for using e as the base instead of trigonometric functions.
  • #1
BucketOfFish
60
1
Hey guys. It's been a few years since I've taken diff-eq and I can't remember how to solve second-order problems like this one:

Find the general solution of
u'' + a^2*u = cos(bx)

I know that if it were homogeneous, I would solve for r^2 + a^2 = 0, and get u = ce^(rx). But for the life of me I can't remember what to do with that cosine on the right side of the equation. Can anyone help?
 
Physics news on Phys.org
  • #2
Welcome to PF!

Hi BucketOfFish! Welcome to PF! :smile:

(try using the X2 icon just above the Reply box :wink:)

You need to find a particular solution (ie any solution to the equation), which you can then add to the general solution to the homogeneous equation.

In this case, try a combination of cos(bx) and sin(bx). :smile:

(if b = ±a, that doesn't work, and you'll need xcos(bx) and xsin(bx))
 
  • #3
Thanks a lot; that really helped!
 
  • #4
I bet you could also use the function:

u''+a2u' = eibx and split the real and imaginary portions with Euler's formula.

I find e easier to work with than sin and cos.
 
  • #5
hi evad1089! :smile:
evad1089 said:
I find e easier to work with than sin and cos.

even when cos is already on the RHS? :wink: …
BucketOfFish said:
u'' + a^2*u = cos(bx)
 

1. What is a second-order nonhomogeneous differential equation?

A second-order nonhomogeneous differential equation is a mathematical expression that relates the second derivative of a function to the function itself, along with other terms that may not be directly related to the function. It is called "nonhomogeneous" because it contains terms that do not depend on the function itself.

2. What is the general form of a second-order nonhomogeneous differential equation?

The general form of a second-order nonhomogeneous differential equation is y'' + p(x)y' + q(x)y = g(x), where y'' is the second derivative of the function y, p(x) and q(x) are functions of x, and g(x) is a function that does not depend on y.

3. How do you solve a second-order nonhomogeneous differential equation?

To solve a second-order nonhomogeneous differential equation, you must first find the general solution to the corresponding homogeneous equation y'' + p(x)y' + q(x)y = 0. Then, you can find a particular solution to the nonhomogeneous equation by using the method of undetermined coefficients or variation of parameters. The general solution of the nonhomogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution.

4. What is the difference between a homogeneous and nonhomogeneous differential equation?

A homogeneous differential equation only contains terms that depend on the function itself, while a nonhomogeneous differential equation contains additional terms that do not depend on the function. This means that the solution to a homogeneous equation will be in the form of y(x) = C1e^r1x + C2e^r2x, where C1 and C2 are constants and r1 and r2 are roots of the characteristic equation. On the other hand, the solution to a nonhomogeneous equation will be in the form of y(x) = yh(x) + yp(x), where yh(x) is the general solution of the corresponding homogeneous equation and yp(x) is a particular solution to the nonhomogeneous equation.

5. What are some real-world applications of second-order nonhomogeneous differential equations?

Second-order nonhomogeneous differential equations are commonly used in physics and engineering to model systems that involve acceleration and forces. Some examples include the motion of a pendulum, the oscillations of a spring, and the behavior of electrical circuits. They are also used in economics to model supply and demand, population growth, and other systems that involve growth or change over time.

Similar threads

Conditions for the undetermined coefficients method to fail with sinusoidal input
    • Calculus and Beyond Homework Help
Replies
12
Views
384
Solving 2nd order ODE in order to get equation for Orbital Trajectory
    • Introductory Physics Homework Help
Replies
4
Views
1K
I Did Math DF's integral calculator make a glaring mistake?
    • Calculus
Replies
8
Views
178
A How to Solve the Fourier Integral in Eq. (27) Involving Position Vectors?
    • Calculus
Replies
10
Views
2K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
791
MHB Solving PDE using laplace transforms
    • Calculus
Replies
4
Views
2K
Need a refresher: 1st order linear diff eq
    • Calculus and Beyond Homework Help
Replies
3
Views
1K
I On vibrating string and differentiating infinite sum
    • Differential Equations
Replies
7
Views
390
Finding ##U## given information about an adiabatic process (Callen)
    • Advanced Physics Homework Help
Replies
5
Views
1K
Looking for book suggestions: see inside.... (physics and diff EQ)
    • Science and Math Textbooks
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top