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"Use the differential operator method to find a fundamental set of solutions [...]"

Could someone please tell me how to do this question so I can do the rest of them in my homework list? I know what the differential operator is from my book but I am not sure how to apply it in a problem.

Any input would be greatly appreciated!

Thanks in advance!

Re: "Use the differential operator method to find a fundamental set of solutions [...

Quote:

Originally Posted by

**s3a** Could someone please tell me how to do this question so I can do the rest of them in my homework list? I know what the differential operator is from my book but I am not sure how to apply it in a problem.

Any input would be greatly appreciated!

Thanks in advance!

Since your question is not very clear I will take a stab at it.

Your equation

$\displaystyle y''+14y'+49y=0$ can be written as

$\displaystyle D^2y+14Dy+49y=0 \iff (D^2+14D+49)y=0 \iff (D+7)^2y=0$

So you need to find everything in the kernel of this operator.

One such vector is $\displaystyle y=Ae^{-7x}$

You can also verify that $\displaystyle y=Bxe^{-7x}$ is in the kernel.

This will gives a spanning set $\displaystyle y_1=e^{-7x}, y_2=xe^{-7x}$