Single equations without initial conditions

First order equations

Equation (1)

ode = (x^4-x^3)*u'[x]+2*x^4*u[x] == x^3/3+C

DSolve[ ode,u,x ]

Equation (2)

ode = -1/2 u'[x] + u[x] == Sin[x]

DSolve[ ode,u,x ]

Equation (3)

ode = y'[x] == y[x]/(y[x] Log[y[x]]+x)

DSolve[ ode,y,x ]

Equation (4)

ode = 2 y[x] y'[x]^2 - 2 x y'[x] - y[x] == 0

DSolve[ ode,y,x ]

Equation (5)

ode = y'[x] + y[x] == y[x]^3 Sin[x]

DSolve[ ode,y,x ]

Equation (6)

ode = y'[x] + P[x] y[x] == Q[x] y[x]^n

The general form of the Bernoulli equation:

DSolve[ ode,y,x ]

Special case:  n = 0

DSolve[ ode /. n->0, y,x ]

Special case:  n = 1

DSolve[ ode /. n->1, y,x ]

DSolve[ ode /. n->Pi, y,x]

And handles n to be complex valued too:

DSolve[ ode /. n->I, y,x ]

Equation (7)

ode= (x^2-1) y'[x]^2 - 2 x y[x]y'[x] + y^2 - 1 == 0

DSolve[ ode,y,x ]

Equation (8)

ode = f[x y'[x]] == g[y'[x]]

DSolve[ ode,y,x ]

Equation (9)

ode = y'[x] == (3 x^2 - y[x]^2 - 7)/(Exp[y[x]]+2 x y[x]+1)

DSolve[ ode,y,x ]

Equation (10)

ode = y'[x] == (2 x^3 y[x] - y[x]^4) / (x^4 - 2 x y[x]^3)

DSolve[ ode,y,x ]

Equation (11)

ode = y'[x] (y'[x] + y[x]) == x (x + y[x])

DSolve[ ode,y,x ]

Factored form of ode11:

odeFactored = (y'[x] + y[x] + x)(y'[x] - x) == 0

DSolve[ odeFactored,y,x ]

Equation (12)

ode = y'[x] == x/(x^2 y[x]^2 + y[x]^5)

DSolve[ ode,y,x ]

Equation (13)

ode = y[x] == 2 x y'[x] - a y'[x]^3

DSolve[ ode,y,x ]

Equation (14)

ode = y[x] == 2 x y'[x] - y'[x]^2

DSolve[ ode,y,x ]

Equation (15)

ode = y'[x] == Exp[x] y[x]^2 - y[x] + Exp[-x]

DSolve[ ode,y,x ]

Equation (16)

ode = y'[x] == y[x]^2 - x y[x] + 1

DSolve[ ode,y,x ]

Equation (17)

ode = y'[x] == (9 x^8 + 1)/(y[x]^2+1)

DSolve[ ode,y,x ]

Equation (18)

ode = y[x] == 2 x y'[x] + y[x] y'[x]^2

DSolve[ ode,y,x ]

Equation (19)

ode = y[x] y'[x] - x y'[x]^2 == x

DSolve[ ode,y,x ]

Second order equations

Equation (20)

ode = y''[x] (a x + b)^2 + 4 y'[x] (a x + b) a + 2 y[x] a^2 == 0

DSolve[ ode,y,x ]

Equation (21)

ode = (x^2-x) u''[x] + (2 x^2+4 x -3) u'[x] +  8 x u[x] == 1

DSolve[ ode,u,x ]

Equation (22)

ode = (x^2-x) w''[x] + (1-2 x^2) w'[x] + (4 x-2) w[x] == 0

DSolve[ ode,w,x ]

Equation (23)

ode = y''[x] - y'[x] == 2 y[x] y'[x]

DSolve[ ode,y,x ]

Equation (24)

ode = y''[x]/y[x] - y'[x]^2/y[x]^2 - 1 + 1/y[x]^3 == 0

DSolve[ ode,y,x ]

Equation (25)

ode = y''[x] + 2 x y'[x] == 2 x

DSolve[ ode,y,x ]

Equation (26)

ode = 2 y[x] y''[x] - y'[x]^2 == 1/3 (y'[x] - x y''[x])^2

DSolve[ ode,y,x ]

Equation (27)

ode = x y''[x] == 2 y[x] y'[x]

DSolve[ ode,y,x ]

Equation (28)

ode = (1-x)(y[x] y''[x] - y'[x]^2) + x^2 y[x]^2 == 0

DSolve[ ode,y,x ]

Equation (29)

ode = x y[x] y''[x] + x y'[x]^2 + y[x] y'[x] == 0

DSolve[ ode,y,x ]

Equation (30)

ode = y''[x]^2 - 2 y'[x] y''[x] - 2 y[x] y'[x] - y[x]^2 == 0

DSolve[ ode,y,x ]

Equation (31)

ode = (x^3/2 - x^2) y''[x] + (2 x^2 - 3 x + 1) y'[x] + 
      (x-1) y[x] == 0

DSolve[ ode,y,x ]

Equation (32)

ode = y''[x] - 2 x y'[x] + 2 y[x] == 3

DSolve[ ode,y,x ]

Equation (33)

ode = Sqrt[x] y''[x] + 2 x y'[x] + 3 y[x] == 0

DSolve[ ode,y,x ]

Equation (34)

ode = x^2 y''[x] + 3 x y'[x] == 1/(y[x]^3 x^4)

DSolve[ ode,y,x ]

Equation (35)

ode = y''[x] - 2/x^2 y[x] == 7 x^4 + 3 x^3

DSolve[ ode,y,x ]

Equation (36)

ode = y''[x] + y[x] ==  Csc[x]

DSolve[ ode,y[x],x ]

Higher order equations

Equation (37)

ode = y'''''''[x]-14 y''''''[x] + 80 y'''''[x] - 
  242 y''''[x] + 419 y'''[x] - 416 y''[x] + 220 y'[x] - 48 y[x] == 0

DSolve[ ode,y,x ]

Equation (38)

ode = y''''[x] - 4/x^2 y''[x] + 8/x^3 y'[x] - 8/x^4 y[x] == 0

DSolve[ ode,y,x ]

Equation (39)

ode = (1+x+x^2) y'''[x] + (3 + 6 x) y''[x] + 6 y'[x] == 6 x

DSolve[ ode,y,x ]

Equation (40)

ode = (y'[x]^2+1) y'''[x] - 3 y'[x] y''[x]^2 == 0

DSolve[ ode,y,x ]

Equation (41)

ode = 3 y''[x] y''''[x] - 5 y'''[x]^2 == 0

DSolve[ ode,y,x ]

Special equations

Equation (42)

ode = y'[t] + a y[t-1] == 0

DSolve[ ode,y,t ]

Equation (43)

ode = D[y[x,a],x] == a y[x,a]

DSolve[ ode,y,x ]

Single equations with initial conditions

Equation (44)

ode = y''''[x] == Sin[x]

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

Equation (45)

ode = x y''[x] + y'[x] + 2 x y[x] == 0

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

Equation (46)

ode = x y'[x]^2 - y[x]^2 + 1 == 0

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

Equation (47)

ode = y''[x] + y[x] y'[x]^3 == 0

DSolve[ {ode, y[0]==0, y'[0]==2},y,x ]

Systems of equations

Equation (48)

DSolve[ {x'[t] == 3 y[t] z[t], y'[t] == 3 x[t] z[t],
         z'[t] == - x[t] y[t]}, {x,y,z},t ]

Equation (49)

DSolve[
  {x'[t] == a[t](y[t]^2-x[t]^2) + 2 b[t] x[t] y[t] + 2 c x[t],
  y'[t] == b[t](y[t]^2-x[t]^2) - 2 a[t] x[t] y[t] + 2 c y[t]},
{x,y},t ]

Equation (50)

DSolve[
  {x'[t] == x[t] (1+Cos[t]/(2+Sin[t])),
   y'[t] == x[t] - y[t]},
{x,y},t ]

Equation (51)

DSolve[ {x'[t] == 9 x[t] + 2 y[t], y'[t] == x[t] + 8 y[t]},
   {x,y}, t ]

Equation (52)

DSolve[ {x'[t] - x[t] + 2 y[t] == 0,
         x''[t] - 2 y'[t] == 2 t - Cos[2 t]},
   {x,y}, t ]

Equation (53)

DSolve[ {y1'[x] == -1/(x (x^2+1)) y1[x] + 1/(x^2 (x^2+1)) y2[x] + 1/x,
         y2'[x] == -x^2/(x^2+1) y1[x] + (2x^2+1)/(x (x^2+1)) y2[x] + 1},
   {y1,y2}, x ]