NOnlinear differential equation method help?

Re your second example: I think the original problem must be to solve

y'(x) + sin[y'(x)/x] = x

Let y' = p merely for the simplicity of writing

x = p + sin(p/x)

The parameter t is introduced with the aim of arranging a single expression for a derivative and we assume, (guess), that dy/dx can be written in the form xt(x) and write

p = xt

x - p = x(1 - t) = sin(p/x) = sin(t)

x = sin(t)/(1 - t)...................(a)

so that now we have

dy/dx = p = [t/(1 - t)]sin(t)....(b)

but we want dy/dt = dy/dx*dx/dt, so we need to calculate dx/dt from (a)

we use quotient rule: [u/v]' = (vu' - uv')/v^2

dx/dt = [(1 - t)cos(t) + sin(t)]/(1 - t)^2...(c)

Multiplying (b) by (c) yields

dy/dt =t sin(t)[(1 - t)cost + sint]/(1 - t)^3

[Note: not [1 - t^3] in denominator}

That explains the part you asked about .

Integration is messy, but just possible (Wolfram) leading to a result

y = f(t) that uses special integral functions Ci(t) and Si(t)

There remains the unexplained part of how to use y = f(t) to get to

y = f(x). Perhaps your teacher will elaborate at your next class.

Regards - Ian