D solve initial and boundary value problems nonlinear ivps and bvps

LogisticEquation = y'[t] == r(1 - (y[t] / K)) * y[t];

DSolve[LogisticEquation, y, t]

{{y -> Function[{t}, C[1]/E^Integrate[-(r*(1 - y[t]/K)), t]]}}

DSolve[{LogisticEquation /. {r -> (1 / 2), K -> 4}, y[0] == 1}, y, t]

{{y -> Function[{t}, E^(Integrate[(-1 - -1/4*y[0])/2, 0] - Integrate[(-1 - -1/4*y[t])/2, t])]}}

sol = DSolve[{LogisticEquation, y[0] == a * K}, y, t]

{{y -> Function[{t}, a*E^(Integrate[-(r*(1 - y[0]/K)), 0] - Integrate[-(r*(1 - y[t]/K)), t])*K]}}

{LogisticEquation, y[0]} /. sol[[1]]//Simplify

{Derivative[1][Function[{t}, a*E^(Integrate[-(r*(1 - y[0]/K)), 0] - Integrate[-(r*(1 - y[t]/K)), t])*K]][t] == -(a^2*E^(2*Integrate[-(r*(1 - y[0]/K)), 0] - 2*Integrate[-(r*(1 - y[t]/K)), t])*K*r) + a*E^(Integrate[-(r*(1 - y[0]/K)), 0] - Integrate[-(r*(1 - y[t]/K)), t])*K*r, a*K}

Plot[Evaluate[Table[y[t] /. sol[[1]] /. {K -> 4, a -> i, r -> (1 / 3)}, {i, 2, 1 / 10, -1 / 3}]], {t, 0, 8}, PlotRange -> All]

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eqn  = y''[x]  - (1 / 2) * (y'[x] ^ 2 / y[x])  + 1 / (2 * y[x]) == 0;

sol = DSolve[{eqn, y[0] == 1, y'[0] == 2}, y, x]

DSolve[{1/(2*y[x]) - Derivative[1][y][x]^2/(2*y[x]) + Derivative[2][y][x] == 0, y[0] == 1, Derivative[1][y][0] == 2}, y, x]

{eqn, y[0], y'[0]} /. sol[[1]]//Simplify

{1/(2*y[x]) - Derivative[1][y][x]^2/(2*y[x]) + Derivative[2][y][x] == 0, y[0], Derivative[1][y][0]} /. {1/(2*y[x]) - Derivative[1][y][x]^2/(2*y[x]) + Derivative[2][y][x] == 0, y[0] == 1, Derivative[1][y][0] == 2}

generalsolution = DSolve[{y''[x] / 2 == y[x] ^ 3 - y[x]}, y[x], x]

DSolve[{Derivative[2][y][x]/2 == -y[x] + y[x]^3}, y[x], x]

(sol = DSolve[{y''[x] / 2 == y[x] ^ 3 - y[x], y[0] == 0, y'[Infinity] == 0}, y[x], x] )//Quiet

DSolve[{Derivative[2][y][x]/2 == -y[x] + y[x]^3, y[0] == 0, Derivative[1][y][Infinity] == 0}, y[x], x]

Plot[{y[x] /. sol[[2]], 1}, {x, -2, 2}, PlotRange -> All]

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