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Wednesday, February 18, 2009

First order and First Degree Differential Equation (Separation of variables)

Separation of variables

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

ƒ1 (x)dx = ƒ2 (y)dy

Solve: y –x dy/dx = a(y2 + dy/dx)

Solution: -

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: -

Solve: dy/dx = x (2 lnx + 1)/siny +y cosy

Solution: -

Given that,
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

Answer: -

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