Find the general solution?

Question

Find the general solution of y'+(1/x-1)y=-2/x.

 

Answer: y=2/x+(c/x)e^x

 

Please show all your work step by step.

Answer 1

y' + (1/x - 1)y = -2/x

 

subtract the second expression from both sides of the equation

 

y' = -(1/x - 1)y - 2/x

 

y' and dy/dx are synonymous

 

dy/dx = -(1/x - 1)y - 2/x

 

Integrate the first & second expression individually.

 

dy/dx = -(1/x - 1)y

 

dy/y = -(1/x - 1)dx

 

Integrate both sides

 

ln y = -ln x + x + c

 

since both sides are equal, use both sides of the equations as exponents with a base e

 

e^(ln y) = e^(-ln x + x + c)

 

y = (ce^x)/x

 

Second part:

 

dy/dx = -2/x

 

dy = (-2/x)dx

 

y = -2ln x + c

 

Complete equation

 

y = (c/x)e^x - 2ln x + Constant 

Answer 2

Sun,

 

Please refer to the other two questions previously answered on this topic and try your hand at them on your own. Be careful to go over it step by step.