# Finding a homogeneous solution

1. Apr 14, 2005

### Naeem

Q. Determine a homogeneous linear differential equation with constant coefficients having having the following solution:

y = C1sin3x + C2cos3x

My idea is to differntiate both sides with respect to x and come up with an equation in dy/dx

what else? can be done......

Is my idea correct.

2. Apr 14, 2005

### estalniath

Hello,

Since the solution is in the form y=ASin3x+BCos3x,
The original equation must have complex roots 3i and -3i.
Thus, a possible solution is d^2y/dx^2+9=0. =)

3. Apr 14, 2005

### Hurkyl

Staff Emeritus
Not every differential equation is a first order equation!

4. Apr 14, 2005

### dextercioby

Pay attention.The characteristic equation is

$$\lambda^{2}+9=0$$

,but the ODE is

$$\frac{d^{2}y}{dx^2}+9y=0$$

Okay?

Daniel.

5. Apr 16, 2005

### Naeem

Can somebody explain how they arrived at $$\lambda^{2}+9=0$$

I know that the two roots are 3i and -3i. I had figured out this already.

6. Apr 16, 2005

### Naeem

Suppose you were given $$\lambda^{2}+9=0$$

How would factor it , in order to find the two values for $$\lambda 7. Apr 16, 2005 ### dextercioby By multiplying $(\lambda-3i)(\lambda+3i)$ and equating it to 0...? Daniel. 8. Apr 16, 2005 ### Naeem Yup I got it thanks!! 9. Apr 23, 2005 ### estalniath Thanks for pointing that out Daniel! I guess that I took the "y" there for granted every time I used the characteristic solution to get the [tex]y_h$$

10. Apr 23, 2005

### HallsofIvy

Staff Emeritus
By the way- this was clearly a simple problem because the given combination was clearly a solution to a linear equation with constant coefficients. It's not always that simple. In general, given a combination of functions with TWO "unknown constants", you form the simplest equation, involving differentials, the eliminates those constants.

If you did NOT recognize y= C1cos(3x)+ C2sin(3x) as coming from &lambda;= 3i and -3i, you could have done this:
Since you are seeking a differential equation: DIFFERENTIATE-
y'= -3 C1 sin(3x)+ 3 C2 cos(3x).
Since there are two unknown constants, DIFFERENTIATE AGAIN-
y"= -9 C1 cos(3x)- 9 C2 sin(3x).

Now do whatever algebraic manipulations you need to eliminate the two constants.

(In this example, of course, just add y" and 3y.)