Identify the type of the differential equation (as linear, homogeneous,exact, separable or Bernoulli) and solve the initial value problemdydt = 2 sin(t) − y, with y(π2)= 3
Question
Identify the type of the differential equation (as linear, homogeneous,exact, separable or Bernoulli) and solve the initial value problemdydt = 2 sin(t) − y, with y(π2)= 3
Solution
The given differential equation is dy/dt = 2sin(t) - y.
This is a first order linear differential equation. It is not homogeneous, exact, separable or Bernoulli.
The general form of a first order linear differential equation is dy/dt + p(t)y = g(t).
Comparing this with the given differential equation, we have p(t) = 1 and g(t) = 2sin(t).
The integrating factor e^(∫p(t) dt) = e^(∫ dt) = e^t.
Multiplying every term of the differential equation by the integrating factor, we get
e^t dy/dt - e^t y = 2e^t sin(t).
This can be written as d/dt (e^t y) = 2e^t sin(t).
Integrating both sides with respect to t, we get
e^t y = -2e^t cos(t) + C.
Therefore, the solution of the differential equation is y = -2 cos(t) + Ce^(-t).
Given that y(π/2) = 3, we can find the constant C.
Substituting t = π/2 and y = 3 into the solution, we get
3 = -2 cos(π/2) + Ce^(-π/2)
=> 3 = Ce^(-π/2)
=> C = 3e^(π/2).
Therefore, the solution of the initial value problem is y = -2 cos(t) + 3e^(π/2 - t).
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