Solving Initial Value Problem with Implicit Solution

[jgiarrusso]

Homework Statement

Solve the initial value problem (y+e^-y)y'=sinx subject to y(pi)=0

Homework Equations

The Attempt at a Solution

I'm not quite sure what to do with this one. I've scanned through my book and could find no similar problems in what we've done so far. I tried to plug in dy/dx for y' and use separation of variables, but then I get stuck unable to solve for y after integrating. I cannot seem to find a way to set this one up to create an integrating factor either. If someone could nudge me in the right direction, I'd greatly appreciate it.

 

[Mark44]

Homework Statement

Solve the initial value problem (y+e^-y)y'=sinx subject to y(pi)=0

Homework Equations

The Attempt at a Solution

I'm not quite sure what to do with this one. I've scanned through my book and could find no similar problems in what we've done so far. I tried to plug in dy/dx for y' and use separation of variables, but then I get stuck unable to solve for y after integrating. I cannot seem to find a way to set this one up to create an integrating factor either. If someone could nudge me in the right direction, I'd greatly appreciate it.

I don't think you're going to be able to solve for y. I ended up with (1/2)y² - e^-y on one side, and a function of x on the other.

Sometimes it's not possible to give the solution as an explicit function of x (ie., as y = f(x)), so the solution is given implicitly.

As long as your equation satisfies the DE (you'll need to use implicit differentiation to solve for dy/dx) and initial condition, you're good.