D solve working with d solve working with dsolvevalue

DSolveValue[y'[x] + y[x] == a Sin[x], y[x], x]

C[1]/E^x + (a*(-Cos[x] + Sin[x]))/2

sol = DSolveValue[{y'[x] + y[x] == a Sin[x], y[0] == 0}, y, x]

Function[{x}, a/(2*E^x) + (a*(-Cos[x] + Sin[x]))/2]

{y'[x] + y[x] == a Sin[x], y[0] == 0} /. y -> sol//Simplify

{(a/E^x + 2*Derivative[1, Function[{x}, a/(2*E^x) + (a*(-Cos[x] + Sin[x]))/2], x] - a*Cos[x] + a*Sin[x])/2 == a*Sin[x], True}

Plot[Table[sol[x] /. {a -> i}, {i, 7}], {x, -2, 10}]

-Graphics-

DSolveValue[{y''[x] + y[x] == 0, y[0] == 0, y'[0] == 1}, y[3], x]

DSolveValue[{y[x] + Derivative[2][y][x] == 0, y[0] == 0, Derivative[1][y][0] == 1}, y[3], x]

DSolveValue[{y''[x] + y[x] == 0, y[0] == 0, y'[0] == 1}, y'[x], x]

DSolveValue[{y[x] + Derivative[2][y][x] == 0, y[0] == 0, Derivative[1][y][0] == 1}, Derivative[1][y][x], x]

DSolveValue[{y'[x] + y[x] == Sin[x], y'[0] ^ 2 == 1}, y[x], x]

DSolveValue[{y[x] + Derivative[1][y][x] == Sin[x], Derivative[1][y][0]^2 == 1}, y[x], x]

DSolve[{y'[x] + y[x] == Sin[x], y'[0] ^ 2 == 1}, y[x], x]

DSolve[{y[x] + Derivative[1][y][x] == Sin[x], Derivative[1][y][0]^2 == 1}, y[x], x]