Need to find y for diferential equaton:
(d^2 y)/(dt^2)=-ay.

I know the solution is y=sin(at), but how to find it from giving equation?

P.S. This diferential equation is for analog computer. http://dcoward.best.vwh.net/analog/argonne.pdf

P.S.S. I bet that analog computer wouldn't calculate 2+8/2-16*3+6/2-5+10-9+13*7*4=321. So about what quantum computer we are talking?

Are you sure that you don't mean the solution is y=sin[sqrt(a)*t] ? The posted solution doesn't satisfy the differential equation (DE):

y = sin(at)
y' = a*cos(at)
y'' = -(a^2)*sin(at)

-a*y = -a*sin(at), so y'' does not equal -a*y. However, y=sin[sqrt(a)*t] does satisfy the DE.

To solve the DE, first make the assumption that the solution is of the the form y = e^[r*t], where r is a root of the the characteristic equation. The characteristic equation is found by plugging the assumed solution into the DE:

y = e^[r*t]
y' = r*e^[r*t]
y'' = r^2*e^[r*t]

DE: y'' + ay = 0

so, r^2*e^[r*t] +a*e^[r*t] = 0

dividing by e^[r*t] yields the characteristic equation: r^2 + a = 0.
The characteristic equation has roots of +i*sqrt(a) and -i*sqrt(a).

Remember that based on our assumption as to the form of the DE's solution, the solution will look like y = e^[r*t]. In order to account for the full range of possible solutions, we have to incorporate both roots into the solution, so modify the form to:

y = A*e^[r1*t] + B*e^[r2*t], where r1 and r2 are the two roots, and A and B are just constants.

Plugging in r1 and r2, we get y = A*e^[i*sqrt(a)*t] + B*e^[-i*sqrt(a)*t]

By using Euler's formula [ e^(i*x) = cos(x) + i*sin(x) ], we can modify this equation to read:

y = A*[cos(sqrt(a)*t) + i*sin(sqrt(a)*t)] + B*[cos(sqrt(a)*t) - i*sin(sqrt(a)*t)]

grouping like terms:

y = (A+B)*cos(sqrt(a)*t) + (A-B)*i*sin(sqrt(a)*t)

If we call (A+B) a constant, say C, and we call (A-B)*i another constant, say D, we can rewrite the equation in even simpler form:

y = C*cos(sqrt(a)*t) + D*sin(sqrt(a)*t)

By using the initial conditions (whatever they may be), this general equation could yield the specific solution y = sin(sqrt(a)*t), which is close to what you posted.

If you haven't taken a differential equations class, I don't know how helpful this explanation will be. Anyway, I hope you got what you wanted, and ask if you'd like any clarification.

I make mistake copying.
The real equation is:
(d^2 y)/(dt^2)=-ya^2 . You can find it in PDF file.
I want to see how need to solve this equation. I hope to solve this equation the solving way will be shorter...

And how to solve this diferential equation
dx/dt+0.1x=5, x(0)=0 ?
Solution is
x(t)=50(1-e^{0.1t}).
http://courses.ece.uiuc.edu/ece486/labs/la...uter_manual.pdf

I recommend sitting an introductory calculus course. Or search for some notes - there will be lots on this subject.