how to solve the differential equations?

Question

I have problems in solving the following differential equations, mainly I do not realize the type of equation:

$$ x'= \cos(t+x), $$ i have no information about $t$ and i have the initial condition : x(0)=pi/2

Answer 1

Using the substitution $u = t+x$ we have $$ u' = x'+1. $$ Substituting this in the equation, we have $$ u' = 1+\cos u, $$ which is a separable equation, leading to $$ \int \frac{du}{1+\cos u} = C + \int dt. $$ It should not be very difficult to show that, evaluating the integrals, on gets $$ \tan \frac{u}{2} = t+C. $$ Substituting back $u=t+x$ and manipulating the result, we have $$ x = 2\arctan (t+C)-t. $$ Applying the initial condition $x(0) = \pi/2$ leads to $C=\tan (\pi/4) = 1$. Then, the solution is $$ x = 2\arctan (t+1)-t. $$