Solving a Non-Linear First Order ODE with Quotient Rule

[jimmythev]

Hello, having a lot of trouble with a dodgy question one of my lecturers has set us before teaching us how to do it, none of my course can seem to work out what to do. The question is:

dy/dx=(x(y+3)+(y+3)²)/x²

where y(1)=4, and x>0

I tried a substitution of z=y/x to eventually give

dy/dx=z²+z+(z+3)/x and am now completely lost. Can anyone help me work out the soln?

 

[CompuChip]

You also need to replace dy/dx by something with just dz/dx (and z and x, but not y).
By the way, note that y/x and (y + 3)/x have the same derivative w.r.t. x, so taking z = (y + 3)/x may be more convenient (although I'm sure it will work out with z = y/x as well).

 

[jimmythev]

If I use z=(y+3)/x I get to dy/dx=z²+z

When I take dz/dx I get -(y+3)/x², so dz=-(y+3)/x² dx,
How do I go about substituting this into the equation so I can integrate wrt z?

 

[Dick]

dz/dx is NOT -(y+3)/x^2. y is a function of x too. You have to use the quotient rule. There will be a dy/dx in the expression for dz/dx. Solve for dy/dx and substitute the result into dy/dx=z^2+z.