Also, for the differential equation:
y'' - y = Sin(2x)sin(x)
How would one find the particular integral?
Second order differential equation:
y'' - y = exp[x]
Solving for the complementary function, y_c:
m^2 - 1 = 0
-> m = 1 and m = 0
:. y_c = A*exp[-1] + B*exp = A*exp[-1] + B
Solving for the particular integral, y_p:
Try y = a*exp[x] -> this is a solution of the homogeneous eqn, so multiply the function by x:
y = ax*exp[x]
y' = a*exp[x] + ax*exp[x]
y'' = a*exp[x] + a*exp[x] + ax*exp[x]
Substituting into y'' - y' = x*exp[x], the equation reduces to:
a + a*x = x -> How to work the constant (alpha) out?