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Differential Equations/A Complex and tedious Simultaneous Differential Equation

Question
Hi. I am doing an exercise questions in specialized book in "Differential equations" By Dennis Zill. I have tried 3 times and my answer doesn't match up to the solution. I hope you can solve it step by step with all the workings. I am quite familiar with procedure of solving operator differential equations second orders. The equations are:

(D^2+5)x -   2y      = 0
-2x     +(D^2 + 2)y = 0

The workings are straight forward and can be found in any textbook.

Solution for the second equation is
y = A cos(sqrt(2) x) + B sin(sqrt(2)x) + x

There is a problem with the first equation: (D^2+5)x - 2y = 0

If the equation is (D^2+5)y - 2y = 0 then the solution would be
y = A e^(sqrt(2)x) + B e^(-sqrt(2)x) + 2.5

Differential Equations

Volunteer

Dr. Nyayapati Swami

Expertise

I can help you in solving first and second order differential equations. Questions must be at the Undergraduate level. Do not expect me to do all your homework.. If you have a homework question with no clues on how to go about, I will only give you some pointers on solving them.

Experience

Ph.D. in Mathematics with more than 20 years of teaching.
In addition to undergraduate calculus, I taught many more advanced subjects like Complex Analysis, General Topology, Numerical Analysis, Operations Research, Graph Theory, Mathematical Analysis, Mathematical Economics, Optimisation Theory.

Education/Credentials
Ph.D. (University of Toledo, USA)