Solving problem analytically

1. Aug 11, 2014

LagrangeEuler

1. The problem statement, all variables and given/known data
Differential equation
$\frac{df}{dx}=\sin f(x)+\cos x$

2. Relevant equations

3. The attempt at a solution
If I integrate equation I will get
$f(x)=\int \sin f(x)dx+\sin x+C$
is there any possibility to solve that analytically?

2. Aug 11, 2014

pasmith

Almost certainly not. There is no general solution method for the ODE $dy/dx = F(x,y)$ subject to $y(x_0) = y_0$ except in special cases, which amount to when $dy/dx = p(x)q(y)$ or there is a change of dependent and/or independent variable which will put the ODE into that form. You then have
$$\int_{y_0}^{y(x)} \frac1{q(s)}\,ds = \int_{x_0}^{x} p(t)\,dt$$
and you then have to ask whether you can (a) do those integrals analytically, and (b) solve the resulting equation for $y(x)$ analytically.

If the answer to either of those questions is "no", then it may be easier to solve the ODE numerically using a suitable method, and if you can't turn your ODE into a separable equation then numerical methods are your only recourse.

3. Aug 11, 2014

Staff: Mentor

Is it possible that there's a typo in the problem statement, and that it should be sin(x) f(x) on the rhs, rather than sin f(x) ?

Chet