We start at the initial value `(0,4)` and calculate the value of the derivative at this point. We have:

`dy/dx=sin(x+y)-e^x`

`=sin(0+4)-e^0`

`=-1.75680249531`

We substitute our starting point and the derivative we just found to obtain the next point along.

`y(x+h)~~y(x)+hf(x,y)`

`y(0.1)~~4+0.1(-1.75680249531)`

`~~3.82431975047`

Now we need to calculate the value of the derivative at this new point `(0.1,3.82431975047)`. We have:

`dy/dx=sin(x+y)-e^x`

`=sin(0.1+3.82431975047)` `-e^0.1`

`=-1.8103864498`

Once again, we substitute our current point and the derivative we just found to obtain the next point along.

`y(x+h)~~y(x)+hf(x,y)`

`y(0.2)~~3.82431975047+` `0.1(-1.8103864498)`

`~~3.64328110549`

We proceed for the required number of steps and obtain these values:

`x` | `y` | `dy/dx` |

0 | 4 | -1.7568024953 |

0.1 | 3.8243197505 | -1.8103864498 |

0.2 | 3.6432811055 | -1.8669109257 |

0.3 | 3.4565900129 | -1.926815173 |

0.4 | 3.2639084956 | -1.9907132334 |

0.5 | 3.0648371723 | -2.0594421065 |

0.6 | 2.8588929616 | -2.1341215746 |

0.7 | 2.6454808042 | -2.2162311734 |

0.8 | 2.4238576868 | -2.3077132045 |

0.9 | 2.1930863664 | -2.4111158431 |

1 | 1.9519747821 |

Here's the graph of this solution.