2nd order non-homogeneneous ODE - how to find PI

In summary, the conversation discusses finding the general and particular solutions of a given ordinary differential equation using the modification rule for particular integrals. The solution involves finding the complementary function and using lambda-notation to determine the roots. The particular solution is then obtained by multiplying the PI by x, and the general solution is the sum of the complementary function and the particular solution.
  • #1
andrew.c
46
0

Homework Statement



Find the general and, if possible, particular solutions of the following ordinary differential equations:

y''+9y=36sin3x
(hint: modification rule for PI)

Homework Equations


Knowledge of ODE's
[tex]y = y_{aux}+y_{particular}[/tex]

The Attempt at a Solution


I get the compementary function;
y''+9y = 0

and then using lambda-notation

[tex]\lambda^2 + 9 = 0[/tex]

therefore

[tex]\lambda = \pm 3i[/tex]
(complex roots)

so
[tex]y_{aux} = A cos 3x + B sin 3x[/tex]

but now how do I get [tex]y_{particular}[/tex]??
 
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  • #2
Notice that '3' is a root in the auxiliary equation.

what is the PI for sin3x? Since '3' is a root, you will need to modify the PI by multiplying it by x.
 
  • #3
PI = particular ?

To expand on what rock.freak667 said, your particular solution should be yp = Axcos(3x) + Bxsin(3x).
 
  • #4
rock.freak667 said:
Notice that '3' is a root in the auxiliary equation

I don't really understand what you mean by this..

However, the PI for sin3x is Acos3x + Bsin3x, i think...
so if you have to multiply by x (don't really understand why?) then the PI would be
x(Acos3x + Bsin3x)?
 
  • #5
Mark44 said:
PI = particular ?

PI = Particular Integral :)
 
  • #6
andrew.c said:
I don't really understand what you mean by this..

However, the PI for sin3x is Acos3x + Bsin3x, i think...
so if you have to multiply by x (don't really understand why?) then the PI would be
x(Acos3x + Bsin3x)?
Not if the complementary solution yc is c1cos3x + c2sin3x, which means that no matter what the values of c1 and c2, yc'' + 9yc = 0. In other words, there is no way you will end up with 36sin3x on the right side.

Your particular solution (I prefer this term to particular integral) must therefore be Axcos3x + Bxsin3x. You need to find the coefficients A and B so that yp'' + 9yp = 36sin(3x).

Your general solution will be y = yc + yp = c1cos3x + c2sin3x + Axcos3x + Bxsin3x, where you will have determined A and B.
 

Related to 2nd order non-homogeneneous ODE - how to find PI

1. What is a 2nd order non-homogenous ODE?

A 2nd order non-homogenous ODE (ordinary differential equation) is an equation that involves a function, its derivatives, and non-zero terms that are not proportional to the function or its derivatives. In simpler terms, it is a mathematical equation that describes a relationship between a function and its derivatives, where the function is not directly proportional to its derivatives.

2. What is PI (particular integral) in the context of 2nd order non-homogenous ODE?

PI, or particular integral, is a solution to a 2nd order non-homogenous ODE that satisfies the non-homogenous terms of the equation. It is a specific solution that is added to the general solution of the equation to satisfy the non-homogenous terms.

3. How do you find the particular integral for a 2nd order non-homogenous ODE?

The particular integral for a 2nd order non-homogenous ODE can be found using the method of undetermined coefficients or variation of parameters. In the method of undetermined coefficients, the particular integral is assumed to have the same form as the non-homogenous terms of the equation. In the variation of parameters method, the particular integral is expressed as a linear combination of the general solution of the corresponding homogeneous equation.

4. What is the importance of finding the particular integral in solving a 2nd order non-homogenous ODE?

The particular integral is necessary to find the complete solution to a 2nd order non-homogenous ODE. Without it, the general solution only satisfies the homogeneous terms of the equation, but does not account for the non-homogenous terms. The particular integral completes the solution and allows for finding the specific solution to the equation.

5. Are there any special cases when finding the particular integral for a 2nd order non-homogenous ODE?

Yes, there are special cases where the particular integral cannot be found using the standard methods. These cases include when the non-homogenous terms are in the form of trigonometric or exponential functions, or when they are in the form of a polynomial multiplied by an exponential function. In these cases, the particular integral is found by assuming a solution that is a combination of the non-homogenous terms.

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