find the general solution of the following first order differential equation.
dx/d(theta)=(xtan(theta))-(cos(theta))
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To find the general solution of the given first-order differential equation, we can use the method of separating variables. Follow these steps:
Step 1: Rearrange the equation to separate the variables, x and θ:
dx/(x tan(θ)) = dθ - cos(θ)
Step 2: Integrate both sides of the equation separately with respect to x and θ:
∫(1/(x tan(θ))) dx = ∫(dθ - cos(θ))
Step 3: Evaluate the integrals on each side:
ln|sec(θ)| = θ - sin(θ) + C
where C is the constant of integration.
Step 4: Rewrite the equation using the properties of logarithms:
sec(θ) = e^(θ - sin(θ) + C)
Step 5: Use trigonometric identities to simplify further:
sec(θ) = e^(θ) * e^(-sin(θ)) * e^(C)
sec(θ) = e^(θ) * (e^(-sin(θ)) * e^(C))
Step 6: Combine the two exponential terms into one constant:
sec(θ) = A * e^(θ)
where A = e^(-sin(θ)) * e^(C) is the new constant.
Step 7: Rewrite sec(θ) in terms of cos(θ):
1/cos(θ) = A * e^(θ)
Step 8: Isolate x on one side of the equation:
x = 1/(A * e^(θ) * cos(θ))
Therefore, the general solution to the given first-order differential equation is:
x = 1/(A * e^(θ) * cos(θ)), where A is a constant.