#=>(d^2 y)/(dx^2) + a^2y = 0#
This is a second-order linear homogenous ODE.
#=>ay'' + by' + cy = 0#
We attempt to find a solution by assuming the solution is of the form of an exponential.
#=>y(x) = e^(gamma x)#
Trying this solution form:
#=> d^2/(dx^2){e^(gamma x)} + a^2e^(gamma x) = 0#
#=> d/(dx){gamma e^(gamma x)} + a^2 e^(gamma x) = 0#
#=> gamma^2 e^(gamma x) + a^2 e^(gamma x) = 0#
#=> (gamma^2 + a^2)e^(gamma x) = 0#
From this result, we need either #gamma^2 + a^2 = 0# or for #e^(gamma x) = 0#. We know that an exponential function cannot be equal to zero, so we need to solve for when #gamma^2 + a^2 = 0#.
#=> gamma^2 + a^2 = 0#
#=> gamma^2 = -a^2#
#=> gamma = +- sqrt(-a^2)#
#=> gamma = +- asqrt(-1)#
#=> gamma = +- ai#
For two distinct (#gamma_1 ne gamma_2#) complex roots of the form:
#=>gamma_1 = alpha + beta i#
#=>gamma_2 = alpha - beta i #
the general solution takes the form:
#=>y(x) = e^(alpha x)(c_1 cos(beta x) + c_2 sin(beta x))#
where #c_1# and #c_2# are arbitrary constants.
In our case, we have #alpha = 0# and #beta = a#. Substituting these values into the general form solution:
#=> y(x) = e^0[c_1 cos(ax) + c_2 sin(ax)]#
Hence the final solution is:
#=> color(green)(y(x) = c_1 cos(ax) + c_2 sin(ax))#
