How do you find all solutions of the differential equation #(d^2y)/(dx^2)=3y#?
1 Answer
# \ \ \ \ \ y = Ae^(sqrt(3)x) + Be^(-sqrt(3)x)#
Where
Explanation:
# (d^2y)/dx^2=3y => (d^2y)/dx^2 + 0dy/dx-3y=0#
This is a Second Order Homogeneous Differential Equation which we solve as follows:
We look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.
# m^2+0m-3 = 0#
# :. m^2-3 = 0#
# :. m = +-sqrt(3)#
Because this has two distinct real solutions
# \ \ \ \ \ y = Ae^(sqrt(3)x) + Be^(-sqrt(3)x)#
Where
-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Verification:
# \ \ \ \ \ \ \ y = Ae^(sqrt(3)x) + Be^(-sqrt(3)x)#
# :. \ y' = Asqrt(3)e^(sqrt(3)x) - Bsqrt(3)e^(-sqrt(3)x) #
# :. y'' = Asqrt(3)sqrt(3)e^(sqrt(3)x) + Bsqrt(3)sqrt(3)e^(-sqrt(3)x) #
# :. y'' = 3Ae^(sqrt(3)x) + 3Be^(-sqrt(3)x) #
Hence;
-+-+-+-+-+-+-+-+-+-+-+-+-+-+