Solve a differential equation

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

{{y[x] -> x^2/2 + C[1]}}

solution = DSolve[y'[x] == x, y[x], x]

{{y[x] -> x^2/2 + C[1]}}

f[x_] = y[x] /. solution[[1]]

x^2/2 + C[1]

f[4]

8 + C[1]

DSolve[{y''[x] == y[x], y[0] == 1, y'[0] == 1}, y[x], x]

{{y[x] -> E^x}}

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

{{y[x] -> E^x*C[1] + C[2]/E^x}}

DSolve[{y'[x] == z[x], z'[x] == -y[x], y[0] == 0, z[0] == 1}, {y[x], z[x]}, x]

DSolve[{Derivative[1][y][x] == z[x], Derivative[1][z][x] == -y[x], y[0] == 0, z[0] == 1}, {y[x], z[x]}, x]

DSolve[{x'[s] == Cos[t[s]], y'[s] == Sin[t[s]], t'[s] == s, x[0] == 0, y[0] == 0, t[0] == 0}, {x[s], y[s], t[s]}, s]

DSolve[{Derivative[1][x][s] == Cos[t[s]], Derivative[1][y][s] == Sin[t[s]], Derivative[1][t][s] == s, x[0] == 0, y[0] == 0, t[0] == 0}, {x[s], y[s], t[s]}, s]

solution  = DSolve[{y''[x] == y[x], y'[0] == 0}, y[x], x]

{{y[x] -> E^x*C[1] + C[2]/E^x}}

g[x_] = y[x] /. solution[[1]]

E^x*C[1] + C[2]/E^x

t[x_] = Table[g[x] /. C[1] -> j, {j, 1, 10}]

{E^x + C[2]/E^x, 2*E^x + C[2]/E^x, 3*E^x + C[2]/E^x, 4*E^x + C[2]/E^x, 5*E^x + C[2]/E^x, 6*E^x + C[2]/E^x, 7*E^x + C[2]/E^x, 8*E^x + C[2]/E^x, 9*E^x + C[2]/E^x, 10*E^x + C[2]/E^x}

Plot[t[x], {x, -2, 2}]

-Graphics-