Differential equation

Solve ( d^2 y(t))/( dt^2) + y(t) = sin(t):
The general solution will be the sum of the complementary solution and particular solution.
Find the complementary solution by solving ( d^2 y(t))/( dt^2) + y(t) = 0:
Assume a solution will be proportional to e^(λ t) for some constant λ.
Substitute y(t) = e^(λ t) into the differential equation:
( d^2 )/( dt^2)(e^(λ t)) + e^(λ t) = 0
Substitute ( d^2 )/( dt^2)(e^(λ t)) = λ^2 e^(λ t):
λ^2 e^(λ t) + e^(λ t) = 0
Factor out e^(λ t):
(λ^2 + 1) e^(λ t) = 0
Since e^(λ t) !=0 for any finite λ, the zeros must come from the polynomial:
λ^2 + 1 = 0
Solve for λ:
λ = i or λ = -i
The roots λ = ± i give y_1(t) = c_1 e^(i t), y_2(t) = c_2 e^(-i t) as solutions, where c_1 and c_2 are arbitrary constants.
The general solution is the sum of the above solutions:
y(t) = y_1(t) + y_2(t) = c_1 e^(i t) + c_2 e^(-i t)
Apply Euler's identity e^(α + i β) = e^α cos(β) + i e^α sin(β):
y(t) = c_1 (cos(t) + i sin(t)) + c_2 (cos(t) - i sin(t))
Regroup terms:
y(t) = (c_1 + c_2) cos(t) + i (c_1 - c_2) sin(t)
Redefine c_1 + c_2 as c_1 and i (c_1 - c_2) as c_2, since these are arbitrary constants:
y(t) = c_1 cos(t) + c_2 sin(t)
Determine the particular solution to ( d^2 y(t))/( dt^2) + y(t) = sin(t) by the method of undetermined coefficients:
The particular solution to ( d^2 y(t))/( dt^2) + y(t) = sin(t) is of the form:
y_p(t) = t (a_1 cos(t) + a_2 sin(t)), where a_1 cos(t) + a_2 sin(t) was multiplied by t to account for sin(t) in the complementary solution.
Solve for the unknown constants a_1 and a_2:
Compute ( d^2 y_p(t))/( dt^2):
( d^2 y_p(t))/( dt^2) = ( d^2 )/( dt^2)(a_1 t cos(t) + a_2 t sin(t))
 = -a_1 t cos(t) - 2 a_1 sin(t) + 2 a_2 cos(t) - a_2 t sin(t)
Substitute the particular solution y_p(t) into the differential equation:
( d^2 y_p(t))/( dt^2) + y_p(t) = sin(t)
-a_1 t cos(t) - 2 a_1 sin(t) + 2 a_2 cos(t) - a_2 t sin(t) + a_1 t cos(t) + a_2 t sin(t) = sin(t)
Simplify:
2 a_2 cos(t) - 2 a_1 sin(t) = sin(t)
Equate the coefficients of cos(t) on both sides of the equation:
2 a_2 = 0
Equate the coefficients of sin(t) on both sides of the equation:
-2 a_1 = 1
Solve the system:
a_1 = -1/2
a_2 = 0
Substitute a_1 and a_2 into y_p(t) = a_2 t sin(t) + a_1 t cos(t):
y_p(t) = -1/2 t cos(t)
The general solution is:
Answer: | y(t) = y_c(t) + y_p(t) = -1/2 t cos(t) + c_1 cos(t) + c_2 sin(t)