Can the Differential Equation df/dx = sin(f(x)) + cos(x) Be Solved Analytically?

[LagrangeEuler]

Homework Statement

Differential equation
$\frac{df}{dx}=\sin f(x)+\cos x$

Homework Equations

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?

 

[pasmith]

Homework Statement

Differential equation
$\frac{df}{dx}=\sin f(x)+\cos x$

Homework Equations

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?

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
<br /> \int_{y_0}^{y(x)} \frac1{q(s)}\,ds = \int_{x_0}^{x} p(t)\,dt<br />
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.

 

[Chestermiller]

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