數學證明題
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證明 consider the third order differential equation y"' - 3y' +2y=0 (1) find the general solution to the differential equation in terms of real functions (2) find the solution to the differential equation which satisfies the initial conditions y(0)= 2 y'(0)=0, and y"(0)=7
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Solve y'''-3y'+2y=0.
The characteristic eq. is t^3-3t+2=0
(t-1)^2 (t+2)=0, t= 1, 1, -2
so, general solution is y=(a+bx)e^x + c e^(-2x)
y(0)=2, then a+c=2
y'(x)=(a+b+bx)e^x - 2c e^(-2x), y'(0)=0, then a+b-2c=0
y"(x)=(a+2b+bx)e^x + 4c e^(-2x), y"(0)=7, then a+2b + 4c=7
so that (a, b, c)=(1, 1, 1)
the sol. is y=(1+x)e^x + e^(-2x)