How to Determine a Differential Equation from a Given Solution?

[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.

 

[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. =)

 

[Hurkyl]

Not every differential equation is a first order equation!

 

[dextercioby]

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. =)

Pay attention.The characteristic equation is

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

,but the ODE is

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

Okay?


Daniel.

 

[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.

 

[Naeem]

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

How would factor it , in order to find the two values for $$\lambda$$

 

[dextercioby]

By multiplying $(\lambda-3i)(\lambda+3i)$ and equating it to 0...?

Daniel.

 

[Naeem]

Yup I got it thanks!

 

[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 $$y_h$$

 

[HallsofIvy]

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 λ= 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.)