Separating Variables: Solving Differential Equations

[Pietair]

Good day,

I have to separate the variables of the formula (dy/dx) + 1 = - (y/x)
so I can determine the solution of the differential equation.

I get:
(dy/dx) + 1 = - (y/x)
(dy/dx) = - (y/x) - 1
(dy) = (- (y/x) - 1)dx

Though I cannot get rid of the y at the side of dx...

 

[foxjwill]

I don't think you can solve it by separation of variables.

 

[Pietair]

Right, thanks. Would it be possible by using Laplace transform?

 

[foxjwill]

possibly, but it would be much easier to use euler's solution to first order linear differential equations.

 

[Pietair]

I already did, but now I want to check the accuracy of the Euler's solution for this differential equation.

 

[HallsofIvy]

I recommend the substitution u= y/x. y= xu so y'= xu'+ u and the equation y'= -y/x- 1 becomes xu'+ u= -u- 1 or x du/dx= -2u-1 which is separable.

 

[rock.freak667]

an integrating factor would work as well.

 

[Pietair]

I recommend the substitution u= y/x. y= xu so y'= xu'+ u and the equation y'= -y/x- 1 becomes xu'+ u= -u- 1 or x du/dx= -2u-1 which is separable.

Thanks!

an integrating factor would work as well.

Thanks, I get:

y(x) = (c/x) - (x/2)

How can I determine the value of the constant now?

 

[HallsofIvy]

Are you asking permission? Certainly if you have some additional condition, you can use that to find c.