Separation of variables
If a differential equation of the first order and first degree is of the form
ƒ1 (x)dx = ƒ2 (y)dy
Given that,
y –x dy/dx = a(y2 + dy/dx)
=> y – ay2 = dy/dx (a + x)
=> y (1 - ay)/dy = (a + x)/dx
=> dx/(a + x) = dy/y (1 - ay)
=> dx/(a + x) = [{a/(1 - ay)} + 1/y] dy
Integrating,
ln(a + x) = - ln(1 - ay) + lny + lnc
=> ln(a + x) = ln{cy/(1 - ay)}
=> a + x = cy/(1 - ay)
Answer: -
dy/dx = x (2 lnx + 1)/siny +y cosy
=>siny +y cosy)dy = (2xlnx + x)dx
Integrating,
ſ sinydy + ſ y cosydy = 2 ſ x lnxdx + ſxdx
=> - cosy + y ſ cosydy - ſ (dy/dy ſ cosydy)dy = 2 lnx ſ xdx - 2 ſ(d lnx/dx ſ xdx)dx + x2/2
=> - cosy +y siny - ſ sinydy = x2 lnx - ſ xdx + x2/2
=> - cosy + y siny + cosy = x2 lnx - x2/2+ x2/2 + c
=> y siny = x2 lnx + c
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