Bob A. answered • 07/28/14
Tutor
4.9
(127)
20 Years Making Science and Maths Understandable and Interesting!
Solving ( dy(t))/( dt)+( d^2 y(t))/( dt^2) = e^t
The general solution is the sum of the complementary solution and particular solution.
Find the complementary solution by solving:
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = 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))+( d)/( dt)(e^(λ t)) = 0
Substitute ( d^2 )/( dt^2)(e^(λ t))
= λ^2 e^(λ t) and ( d)/( dt)(e^(λ t)) = λ e^(λ t)
λ^2 e^(λ t)+λ e^(λ t) = 0
Factor out e^(λ t):
(λ^2+λ) e^(λ t) = 0
Since e^(λ t) !=0 for any finite λ,
the zeros must come from the polynomial:
λ^2+λ = 0
Factor:
λ (λ+1) = 0
Solve for λ:
λ = -1 or λ = 0
The root λ = -1 gives y_1(t)
= c_1 e^(-t) as a solution, where c_1 is an arbitrary constant.
The root λ = 0 gives y_2(t)
= c_2 as a solution, where c_2 is an arbitrary constant.
The general solution is the sum of the above solutions:
y(t) = y_1(t)+y_2(t) = c_1/e^t+c_2
Determine the particular solution to
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = e^t
by the method of undetermined coefficients:
The particular solution to
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = e^t is of the form:
y_p(t) = a_1 e^t
Solve for the unknown constant a_1:
Compute ( dy_p(t))/( dt):
( dy_p(t))/( dt) = ( d)/( dt)(a_1 e^t)
= a_1 e^t
Compute ( d^2 y_p(t))/( dt^2):
( d^2 y_p(t))/( dt^2) = ( d^2 )/( dt^2)(a_1 e^t)
= a_1 e^t
Substitute the particular solution y_p(t) into the differential equation:
( d^2 y_p(t))/( dt^2)+( dy_p(t))/( dt) = e^t
a_1 e^t+a_1 e^t = e^t
Simplify:
2 a_1 e^t = e^t
Equate the coefficients of e^t on both sides of the equation:
2 a_1 = 1
Solve the equation:
a_1 = 1/2
Substitute a_1 into y_p(t) = a_1 e^t:
y_p(t) = e^t/2
The general solution is:
y(t) = y_c(t)+y_p(t) = e^t/2+c_1/e^t+c_2
The general solution is the sum of the complementary solution and particular solution.
Find the complementary solution by solving:
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = 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))+( d)/( dt)(e^(λ t)) = 0
Substitute ( d^2 )/( dt^2)(e^(λ t))
= λ^2 e^(λ t) and ( d)/( dt)(e^(λ t)) = λ e^(λ t)
λ^2 e^(λ t)+λ e^(λ t) = 0
Factor out e^(λ t):
(λ^2+λ) e^(λ t) = 0
Since e^(λ t) !=0 for any finite λ,
the zeros must come from the polynomial:
λ^2+λ = 0
Factor:
λ (λ+1) = 0
Solve for λ:
λ = -1 or λ = 0
The root λ = -1 gives y_1(t)
= c_1 e^(-t) as a solution, where c_1 is an arbitrary constant.
The root λ = 0 gives y_2(t)
= c_2 as a solution, where c_2 is an arbitrary constant.
The general solution is the sum of the above solutions:
y(t) = y_1(t)+y_2(t) = c_1/e^t+c_2
Determine the particular solution to
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = e^t
by the method of undetermined coefficients:
The particular solution to
( d^2 y(t))/( dt^2)+( dy(t))/( dt) = e^t is of the form:
y_p(t) = a_1 e^t
Solve for the unknown constant a_1:
Compute ( dy_p(t))/( dt):
( dy_p(t))/( dt) = ( d)/( dt)(a_1 e^t)
= a_1 e^t
Compute ( d^2 y_p(t))/( dt^2):
( d^2 y_p(t))/( dt^2) = ( d^2 )/( dt^2)(a_1 e^t)
= a_1 e^t
Substitute the particular solution y_p(t) into the differential equation:
( d^2 y_p(t))/( dt^2)+( dy_p(t))/( dt) = e^t
a_1 e^t+a_1 e^t = e^t
Simplify:
2 a_1 e^t = e^t
Equate the coefficients of e^t on both sides of the equation:
2 a_1 = 1
Solve the equation:
a_1 = 1/2
Substitute a_1 into y_p(t) = a_1 e^t:
y_p(t) = e^t/2
The general solution is:
y(t) = y_c(t)+y_p(t) = e^t/2+c_1/e^t+c_2
