Separation of variables

If a differential equation of the first order and first degree is of the form

ƒ_{1} (x)dx = ƒ_{2} (y)dy

^{2} + dy/dx)

Given that,

y –x dy/dx = a(y^{2} + dy/dx)

=> y – ay^{2} = 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 + x^{2}/2

=> - cosy +y siny - ſ sinydy = x^{2} lnx - ſ xdx + x^{2}/2

=> - cosy + y siny + cosy = x^{2} lnx - x^{2}/2+ x^{2}/2 + c

=> y siny = x^{2} lnx + c

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