Lösungen

Remove["Global`*"]

1

DSolve[y'[x] == x y[x]^3, y[x], x ]

DSolve[{y'[x] == x y[x]^3, y[1] == 1}, y[x], 
      x ]

DSolve[{y'[x]==x y[x]^3,y[1]==1},y,x ]

u[x_, c_] := Re[PowerExpand[(c /x)^(1/3)]]; u[x, c]

u[x_, c_] := Abs[(c /x)]^(1/3) Sign[x c]; u[x, c]

g1 = 
        Plot[Evaluate[Table[u[x, c^3], {c, -5, 5, 0.3}]], {x, -2, 2}, 
          PlotRange -> {-2, 2}];

h[x_]:=ToRadicals[Table[Root[1/x^3 - c, 1],{c,1,5}]];

1/ToRadicals[Table[Root[x^3 - c, 1],{c,1,5}]]//N

ToRadicals[Table[Root[x^3 - c, 1],{c,1,5}]]//N

?Root

<< Graphics`PlotField`

g2=PlotVectorField[{1, x y^3},{x,-2,2,0.2},{y,-2,2,0.2}];

Show[g1, g2];

2

DSolve[{y'[x]==Cos[x]/y[x]},y,x ]

DSolve[{y'[x]==Cos[x]/y[x],y[0]==4},y[x],x ]

DSolve[{y'[x] == Cos[x]/y[x], y[0] == 4}, y, 
      x ]

3

a

DSolve[{y[x] + 4  x +  y'[x] x == 0}, y[x], x ] // 
      Simplify

b

y = - 2 x

4

Remove["Global`*"]

DSolve[{y''[x] - y'[x] +  y[x]  == E^(-x)}, 
        y[x], x ] // Simplify

DSolve[{y''[x] - y'[x] +  y[x]  == E^(-x), 
          y[0] == 0, y'[0] == 0}, y[x], x ] // 
      Simplify

solv=DSolve[{y''[x]-y'[x]+ y[x] ==E^(-x),y[0]==0,y'[0]==0},y,x ]//Simplify//Flatten

Wichtig: Am Schluss steht ....y,x ]//Simplify//Flatten und nicht ....y[x],x ]//Simplify//Flatten

y=y/.solv;
y[z]

y[1]

y[1]//N

pl=Plot[y[z],{z,0,10}];