find the general solution of the following first order differential equation.
dx/d(theta)=(xtan(theta))-(cos(theta))
To find the general solution of the first-order differential equation:
(dx/dθ) = xtan(θ) - cos(θ),
we can use the method known as separation of variables. The idea is to separate the variables x and θ so that all terms involving x are on one side and all terms involving θ are on the other side.
Step 1: Rearrange the equation to separate the variables:
(dx/x) = (xtan(θ) - cos(θ)) dθ.
Step 2: Integrate both sides with respect to the corresponding variables:
∫(dx/x) = ∫(xtan(θ) - cos(θ)) dθ.
Step 3: Evaluate the integrals:
ln|x| = (1/2)x^2 - ln|sec(θ)| + C,
where C is the constant of integration.
Step 4: Solve for x:
ln|x| = (1/2)x^2 - ln|sec(θ)| + C.
Using properties of logarithms and exponentiating both sides:
|x| = e^[(1/2)x^2 - ln|sec(θ)| + C],
|x| = e^(1/2)x^2 / sec(θ)e^C,
|x| = Ce^(1/2)x^2 / sec(θ),
where C = ± e^C.
Hence, the general solution of the given first-order differential equation is:
x = ± Ce^(1/2)x^2 / sec(θ).
Note: The constant C takes different values for different initial conditions or boundary conditions.