Solve the initial value problem
(dy/dt)+2y=15 sin(t)+15 cos(t)
with y(0)=3
To solve the initial value problem, follow these steps:
Step 1: Find the general solution of the homogeneous equation.
The homogeneous equation is (dy/dt) + 2y = 0. It is a first-order linear homogeneous ordinary differential equation. The general form of the solution is y(t) = Ce^(-2t), where C is an arbitrary constant.
Step 2: Find a particular solution for the non-homogeneous equation.
To find a particular solution for the non-homogeneous equation, we can use the method of undetermined coefficients. We guess the form of the particular solution and adjust the parameters until we get a valid solution.
In this case, the right-hand side of the equation is 15 sin(t) + 15 cos(t). We can guess a particular solution of the form y_p(t) = A sin(t) + B cos(t). We substitute this guess into the equation and solve for A and B.
(dy/dt) + 2y = A cos(t) - B sin(t) + 2(A sin(t) + B cos(t))
= (A + 2B) cos(t) + (2A - B) sin(t)
Comparing the coefficients of cos(t) and sin(t) on both sides of the equation, we get:
A + 2B = 0
2A - B = 15
Solving these equations simultaneously, we find A = 10 and B = -5. Therefore, the particular solution is y_p(t) = 10 sin(t) - 5 cos(t).
Step 3: Combine the general and particular solutions.
The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution. Therefore, the general solution is y(t) = Ce^(-2t) + 10 sin(t) - 5 cos(t).
Step 4: Apply the initial condition to find the value of the constant C.
We are given that y(0) = 3. Substituting t = 0 and y = 3 into the general solution, we have:
3 = Ce^(0) + 10 sin(0) - 5 cos(0)
3 = C - 5
Solving for C, we find C = 8.
Step 5: Final solution.
Substituting C = 8 into the general solution yields:
y(t) = 8e^(-2t) + 10 sin(t) - 5 cos(t)
Therefore, the solution to the initial value problem is y(t) = 8e^(-2t) + 10 sin(t) - 5 cos(t), with the initial condition y(0) = 3.
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