Solve the initial value problem
(dx/dt)+2x=cos(2t)
with x(0)=5
To solve the initial value problem (IVP), we will use an integrating factor and then integrate both sides of the equation.
Step 1: Find the integrating factor.
The integrating factor, denoted by μ(t), is given by the exponential of the integral of the coefficient of x, which in this case is 2. Therefore, μ(t) = e^(∫2 dt) = e^(2t).
Step 2: Multiply both sides of the differential equation by the integrating factor.
e^(2t) * (dx/dt) + 2e^(2t)x = e^(2t) * cos(2t).
Step 3: Simplify the left side using the product rule.
(d/dt)(e^(2t) * x) = e^(2t) * cos(2t).
Step 4: Integrate both sides with respect to t.
∫(d/dt)(e^(2t) * x) dt = ∫e^(2t) * cos(2t) dt.
By applying the fundamental theorem of calculus, we can integrate the left side of the equation:
e^(2t) * x = ∫e^(2t) * cos(2t) dt.
We can find this integral using integration techniques or a table of integrals. The integral of e^(2t) * cos(2t) can be obtained as a linear combination of sine and cosine functions.
Let's assume the integral is of the form A * sin(2t) + B * cos(2t), where A and B are constants. We then differentiate this expression and set it to e^(2t) * cos(2t). By comparing coefficients, we can solve for A and B.
Differentiating the assumed integral with respect to t, we get:
(d/dt)(A * sin(2t) + B * cos(2t)) = 2A * cos(2t) - 2B * sin(2t).
Comparing this with e^(2t) * cos(2t), we have the following equations:
2A = 1 and -2B = 1.
Solving these equations, we find A = 1/2 and B = -1/2.
Therefore, the integral of e^(2t) * cos(2t) is (1/2) * sin(2t) - (1/2) * cos(2t).
Substituting this result back into the equation, we have:
e^(2t) * x = (1/2) * sin(2t) - (1/2) * cos(2t) + C,
where C is the constant of integration.
Step 5: Solve for x.
To find x, divide both sides of the equation by e^(2t):
x = [(1/2) * sin(2t) - (1/2) * cos(2t) + C] / e^(2t).
Step 6: Apply the initial condition.
We are given that x(0) = 5. Substituting t = 0 and x = 5 in the equation, we can solve for C:
5 = [(1/2) * sin(2 * 0) - (1/2) * cos(2 * 0) + C] / e^(2 * 0),
5 = [0 - 1/2 + C] / 1,
5 = (-1/2 + C).
Simplifying, we have:
5 + 1/2 = C,
C = 11/2.
Therefore, the particular solution to the IVP is:
x = [(1/2) * sin(2t) - (1/2) * cos(2t) + 11/2] / e^(2t).
chris or lisa or whoever --
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