Order = 3 and Degree = 1
The order of a differential equation is determined by the highest-order derivative; the degree is determined by the highest power on a variable. The higher the order of the differential equation, the more arbitrary constants need to be added to the general solution.

The order of a differential equation representing a family of curves is same as the number of arbitrary constants present in the equation corresponding to the family of curves.

x³ dx + (y + 1)dy = 0
=>d(x⁴ /4) + d((y + 1)³ /3) = 0
=>d(x⁴ /4 + ((y + 1)³ /3)) = 0
=>(x⁴ /4 + ((y + 1)³ /3)) = constt
=>12 *(x⁴ /4 + ((y + 1)³ /3)) = 12 * constt
=>3x⁴ + 4(y + 1)³ = constt

(x −y)dy/dx = x + 2y
⇒dy/dx = (x + 2y)/(x −y)
F (x,y) = (x + 2y)/(x −y)
F(Ax, Ay) = (Ax + 2Ay)/(Ax −Ay) 
= A(x + 2y)/A(x −y) 
= (x + 2y)/(x −y) = F (x,y)
Hence, the equation is homogenous.

xdy/dx = -coty
dy/dx = - coty/x
dy/coty = - dx/x
tany dy = - dx/x  
∫(siny/cosy)dy = - ∫dx/x …………..(1)
Put t = cosy
dt = -sinydy
Put the value of dt in eq(1)
-∫dt/t = -∫dx/x
log |t| = log |x| + log c
log |cosy| = log(c.|x|)
|cos y| = c.|x|....................(2)
As y = π/4, x = (2)^½
cos π/4 = c*(2)^½
1/(2)^½  = c*(2)^½  
c = ½
Put the value of ‘c ’in eq(2)
cos y = x/2   =>x = 2cosy

dy/dx = e^ax cos by  
∫dy/cos by = ∫e^ax dx
∫sec by dy = ∫e^ax dx
= (log| sec by + tan by|)/b = e^ax /a + c
= a(log| sec by + tan by|) = be^ax + c

dy/dx = x logx
=>∫dy = ∫x logx dx
y = logx . x² /2 - ∫1/x . x² /2 dx + c
y = x² /2 log x -½∫x dx + c
y = x² /2 log x - ½. x² /2 + c
y = x² /2 log x - x² /4 + c

 d² y/dx² = xe^x
d² y/dx² =  ∫xe^x
d/dx(dy/dx) = [xe^x ]
Integrating, we get
dy/dx = (x-1)e^x + c1
Again integrating, y = (x-2)e^x + c1x + c2

 e^dy/dx = x + 1
⇒dy/dx = log(x+1)
⇒dy = log(x+1)dx
Integrating both sides, we get
⇒∫dy =  ∫log(x+1)dx
y = log(x + 1)  ∫1dx - ∫[d/dx{log(x + 1)} ∫1dx]dx
y = xlog(x + 1) -  ∫1/(x+1)xdx
y = xlog(x + 1) -  ∫(1 - 1/(x + 1)dx
y = xlog(x + 1) -  ∫dx +  ∫1/(x+1)dx
y = xlog(x + 1) - x + log|x + 1| + c
y = (x + 1)log|x + 1| - x + c
Here c is 3
y = (x + 1)log|x + 1| - x + 3