Solve dy/dx + y = sin(exp(x))

Is this differential equation separable? I attempted it to solve it by integrating factor but ended up with exp(x).y = int(sin(exp(x)),x) which I believe the right hand side is impossible to integrate.

Solve for y(x).

ps: int(sin(exp(x)),x) means indefinite integral of sin(exp(x)) with respect to x

The IF is e^(S(1dx)=e^x and multiplying by the IF gives
[ye^x}' = (e^x)sin(e^x) and integrating gives ye^x = - cos(e^x)+c
so y= - e^(-x)cos(e^x)+ce^(-x)

You made a slip in your line 2, but nearly right.

first its a first order differential

so

the integrating factor is u = int exp(1) dx

this means

u = exp(x)

udy/dx + uy = u sin(exp(x))

the left side is simply

d(uy)/dx = u dy/dx + u y

so

d(uy)/dx = u sin(exp(x))

seperate the variables


d(uy) = u sin(exp(x)) dx

integrate both sides

u y = int ( u sin(exp(x)) dx

this should give you enought information to solve it hope it helps