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x.diff(x)*(y)=x+e^y equation

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Numerical solution:

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The solution

You have entered [src] 
//x  for 0 = 1\           
||            |          y
|<1  for 1 = 1|*y = x + E 
||            |           
\\0  otherwise/
y({xfor 0=11for 1=10otherwise)=ey+xy \left(\begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases}\right) = e^{y} + x
Simplify
The graph
Sum and product of roots [src] 
sum
    / /  x\\     /    / /  x\\        \        
- re\W\-e // + I*\- im\W\-e // + im(x)/ + re(x)
i(im⁡(x)−im⁡(W(−ex)))+re⁡(x)−re⁡(W(−ex))i \left(\operatorname{im}{\left(x\right)} - \operatorname{im}{\left(W\left(- e^{x}\right)\right)}\right) + \operatorname{re}{\left(x\right)} - \operatorname{re}{\left(W\left(- e^{x}\right)\right)} 
=
    / /  x\\     /    / /  x\\        \        
- re\W\-e // + I*\- im\W\-e // + im(x)/ + re(x)
i(im⁡(x)−im⁡(W(−ex)))+re⁡(x)−re⁡(W(−ex))i \left(\operatorname{im}{\left(x\right)} - \operatorname{im}{\left(W\left(- e^{x}\right)\right)}\right) + \operatorname{re}{\left(x\right)} - \operatorname{re}{\left(W\left(- e^{x}\right)\right)} 
product
    / /  x\\     /    / /  x\\        \        
- re\W\-e // + I*\- im\W\-e // + im(x)/ + re(x)
i(im⁡(x)−im⁡(W(−ex)))+re⁡(x)−re⁡(W(−ex))i \left(\operatorname{im}{\left(x\right)} - \operatorname{im}{\left(W\left(- e^{x}\right)\right)}\right) + \operatorname{re}{\left(x\right)} - \operatorname{re}{\left(W\left(- e^{x}\right)\right)} 
=
    / /  x\\     /    / /  x\\        \        
- re\W\-e // + I*\- im\W\-e // + im(x)/ + re(x)
i(im⁡(x)−im⁡(W(−ex)))+re⁡(x)−re⁡(W(−ex))i \left(\operatorname{im}{\left(x\right)} - \operatorname{im}{\left(W\left(- e^{x}\right)\right)}\right) + \operatorname{re}{\left(x\right)} - \operatorname{re}{\left(W\left(- e^{x}\right)\right)} 
-re(LambertW(-exp(x))) + i*(-im(LambertW(-exp(x))) + im(x)) + re(x)
Rapid solution [src] 
         / /  x\\     /    / /  x\\        \        
y1 = - re\W\-e // + I*\- im\W\-e // + im(x)/ + re(x)
y1=i(im⁡(x)−im⁡(W(−ex)))+re⁡(x)−re⁡(W(−ex))y_{1} = i \left(\operatorname{im}{\left(x\right)} - \operatorname{im}{\left(W\left(- e^{x}\right)\right)}\right) + \operatorname{re}{\left(x\right)} - \operatorname{re}{\left(W\left(- e^{x}\right)\right)} 
y1 = i*(im(x) - im(LambertW(-exp(x)))) + re(x) - re(LambertW(-exp(x)))
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