How do you solve the equation (x+y-1)dx+(y-x-5)dy=0?

[skyturnred]

Homework Statement

Solve (x+y-1)dx+(y-x-5)dy=0

Homework Equations

The Attempt at a Solution

x=u+h
y=v+k

Therefore

h=-2 and k=3

Therefore

(u+v)du+(v-u)dv=0

[itex]\frac{dv}{du}[/itex]=[itex]\frac{u+v}{u-v}[/itex]

This is where I'm stuck. I can't solve the equation above. I have a feeling I need to rearrange so that I have a function of [itex]\frac{v}{u}[/itex] on the right hand side, but I am unable to rearrange it to attain this.

Any suggestions?

Thank-you

 

[CAF123]

The purpose of the initial substitution was to find an appropriate h,k such that the constant terms in the eqn in the OP vanished and hence arrive at precisely the form you now have. Define another variable ##f = \frac{v}{u}## after rearranging the RHS of your eqn.

 

[skyturnred]

The purpose of the initial substitution was to find an appropriate h,k such that the constant terms in the eqn in the OP vanished and hence arrive at precisely the form you now have. Define another variable ##f = \frac{v}{u}## after rearranging the RHS of your eqn.

OK, I have tried this already. I still get stuck however.

g=[itex]\frac{v}{u}[/itex]

v=gu

[itex]\frac{dv}{du}[/itex]=u[itex]\frac{dg}{du}[/itex]+g

so when you sub this into [itex]\frac{dv}{du}[/itex]=[itex]\frac{u+v}{u-v}[/itex] you get

u[itex]\frac{dg}{du}[/itex]+g=[itex]\frac{\frac{v}{g}+gu}{\frac{v}{g}-gu}[/itex]

And I'm still stuck. This seemed even more complicated than before the substitution.

 

[CAF123]

OK, I have tried this already. I still get stuck however.

g=[itex]\frac{v}{u}[/itex]

v=gu

[itex]\frac{dv}{du}[/itex]=u[itex]\frac{dg}{du}[/itex]+g

so when you sub this into [itex]\frac{dv}{du}[/itex]=[itex]\frac{u+v}{u-v}[/itex] you get

u[itex]\frac{dg}{du}[/itex]+g=[itex]\frac{\frac{v}{g}+gu}{\frac{v}{g}-gu}[/itex]

And I'm still stuck. This seemed even more complicated than before the substitution.

You have the eqn $$\frac{dv}{du} = \frac{u+v}{u-v} = \frac{1 + \frac{v}{u}}{1 - \frac{v}{u}}$$ You correctly computed dv/du to obtain ##u\frac{dg}{du} + g## but the RHS of the eqn that you wrote above is incorrect. You want a RHS to be entirely in terms of g and u so you can apply separation of variables.

 

[skyturnred]

You have the eqn $$\frac{dv}{du} = \frac{u+v}{u-v} = \frac{1 + \frac{v}{u}}{1 - \frac{v}{u}}$$ You correctly computed dv/du to obtain ##u\frac{dg}{du} + g## but the RHS of the eqn that you wrote above is incorrect. You want a RHS to be entirely in terms of g and u so you can apply separation of variables.

Thank-you. I will be able to solve it from here on.

My biggest issue was in realizing that [itex]\frac{u+v}{u-v}[/itex]=[itex]\frac{1+\frac{v}{u}}{1-\frac{v}{u}}[/itex]

After this I rearranged to

[itex]\frac{du}{u}[/itex]=[itex]\frac{1-g}{1+g^{2}}[/itex]dg

I should be able to solve this using techniques of integration.

Thank-you once again