Try,using NDSolve:
ClearAll["Global`*"]; Remove["Global`*"];
sol = NDSolve[{x'[t] == t, y'[t] == t^2, z'[t] == 3*t, x[0] == 1,
y[0] == 0, z[0] == 0}, {x, y, z}, {t, 0, 10}];
ParametricPlot3D[{x[t], y[t], z[t]} /. sol, {t, 0, 10},
PlotRange -> All, AspectRatio -> 1/2,
AxesLabel -> {"x[t]", "y[t]", "z[t]"}]
a = 1/10;
sol2 = NDSolve[{x'[
t] == (-Sin[t]*(1 + a^2 + t^2) - 2*t*Cos[t])/(1 + a^2 + t^2)^2,
y'[t] == (Cos[t]*(1 + a^2 + t^2) - 2*t*Sin[t])/(1 + a^2 + t^2)^2,
z'[t] == a*(1 + a^2 - t^2)/(1 + a^2 + t^2)^2, x[0] == 1, y[0] == 0,
z[0] == 0}, {x, y, z}, {t, 0, 10}];
ParametricPlot3D[{x[t], y[t], z[t]} /. sol2, {t, 0, 10},
PlotRange -> All, AspectRatio -> 1,
AxesLabel -> {"x[t]", "y[t]", "z[t]"}]
Or using ParametricNDSolve:
ClearAll["Global`*"]; Remove["Global`*"];
sol3 = ParametricNDSolve[{x'[
t] == (-Sin[t]*(1 + a^2 + t^2) - 2*t*Cos[t])/(1 + a^2 + t^2)^2,
y'[t] == (Cos[t]*(1 + a^2 + t^2) - 2*t*Sin[t])/(1 + a^2 + t^2)^2,
z'[t] == a*(1 + a^2 - t^2)/(1 + a^2 + t^2)^2, x[0] == 1,
y[0] == 0, z[0] == 0}, {x, y, z}, {t, 0, 10}, {a}];
ParametricPlot3D[{x[a][t], y[a][t], z[a][t]} /. sol3 /. a -> 1/10, {t,0, 10}, PlotRange -> All, AspectRatio -> 1, AxesLabel -> {"x[t]", "y[t]", "z[t]"}]