The Solution of Differential Equation d²y/dt²=0
Answers
What is the complete solution of D²y/Dx²+y=0?
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This is just a homogeneous second-order, linear differential equation with constant coefficients. The correct terminology is “general solution”. You can write it one of two ways:
i. y=C1⋅eix+C2⋅e−ix
or
ii. y=C3⋅cos(x)+C4⋅sin(x)
If you have real initial conditions, the second way is the easiest starting point. With real initial conditions, you can find linear combinations of the constants in i. that give you the constants in ii. The complex exponentials would combine to give the sine and cosine, but you can just start with the second version and skip a lot of steps.
If for some reason your initial values were complex, you might just keep the solution in the first form.
By the way, you asked for a “complete” solution. If you mean that you also need to find the constants, then you need 2 initial conditions, usually the value of y and y’ at some particular x-value. When you find the constants the end result is also called a “unique” solution.