Nonhomogeneous 2nd order DE

Thread starter Sean77771
Start date Oct 22, 2008
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2nd order Nonhomogeneous

In summary, a nonhomogeneous 2nd order differential equation is an equation that involves the second derivative of a function, as well as the function itself, and also includes a non-zero function on the right side of the equation. To solve this type of equation, one can use the method of undetermined coefficients or the method of variation of parameters. A homogeneous 2nd order differential equation does not have a non-zero function on the right side, while a nonhomogeneous one does, making it more difficult to solve. Nonhomogeneous 2nd order differential equations have real-world applications in modeling physical systems and are commonly used in engineering, economics, and physics. To check if a solution to a nonhomogeneous 2nd order

Oct 22, 2008

Sean77771

Homework Statement

y'' + 9y = 2x²e^3x + 5

Homework Equations

N/A

The Attempt at a Solution

I think the complementary solution y_c = c₁cos(3x) + c₂sin(3x).

If not for that little +5 at the end of the right hand side, I'm pretty sure I could solve it. But I don't know how to include it in my solution.

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Oct 22, 2008

HallsofIvy

Science Advisor

Homework Helper

Try adding a constant to your specific solution.

1. What is a nonhomogeneous 2nd order differential equation?

A nonhomogeneous 2nd order differential equation is an equation that involves the second derivative of a function, as well as the function itself, and also includes a non-zero function on the right side of the equation. This non-zero function is known as the nonhomogeneous term.

2. How do you solve a nonhomogeneous 2nd order differential equation?

To solve a nonhomogeneous 2nd order differential equation, you can use the method of undetermined coefficients or the method of variation of parameters. Both methods involve finding a particular solution to the equation and then adding it to the general solution of the corresponding homogeneous equation.

3. What is the difference between a homogeneous and nonhomogeneous 2nd order differential equation?

A homogeneous 2nd order differential equation does not have a non-zero function on the right side of the equation, while a nonhomogeneous 2nd order differential equation does. This non-zero function can make the equation more difficult to solve, as it adds an extra variable to consider.

4. What are some real-world applications of nonhomogeneous 2nd order differential equations?

Nonhomogeneous 2nd order differential equations can be used to model various physical systems, such as the motion of a swinging pendulum or the growth of a population. They are also commonly used in engineering, economics, and physics to describe the behavior of systems and predict future outcomes.

5. How do you check if a solution to a nonhomogeneous 2nd order differential equation is correct?

To check if a solution to a nonhomogeneous 2nd order differential equation is correct, you can substitute the solution into the original equation and see if it satisfies the equation. Additionally, you can also check if the solution satisfies any initial conditions given in the problem.