Verify that the general solution f(x)=c_1sin(ix)+c_2cos(ix) satisfies the ODE f''(x)=f(x).
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Verify that the general solution f(x)=c_1sin(ix)+c_2cos(ix) satisfies the ODE f''(x)=f(x).
We show that f(x)=c_1sin(ix)+c_2cos(ix) is the general solution of f''(x)=f(x) by taking successive derivatives of f(x).
In this problem, i is the imaginary unit, so each time we differentiate f(x)=c_1sin(ix)+c_2cos(ix), the chain rule brings out a factor of i. After the second derivative, we have two factors of i or i^2 which is -1. Then we recognize that the second derivative of the given function is equal to the original function, so the function satisfies the differential equation f''(x)=f(x).
Since the function has two arbitrary constants in it and satisfies the 2nd order ODE, it is a general solution of the differential equation f''(x)=f(x).
