Solving D.eq. with Homogeneous Method: Tips and Tricks for Beginners

[em919]

I need some help to solve this D.eq. I've tried the homogeneous method but I didn't work!

Which method can I use?

y'=(-3x+4y-18)/(2x-y+7)

 

[Metaleer]

Hi, em919! ;)

First of all, you need to solve the simultaneous equations

$$-3x+4y-18 = 0,$$

$$2x-y+7 = 0.$$​

If the solution to this set of simultaneous equations is $$(x_1, y_1)$$, you need to make a couple of changes of variables: $$t = x + x_1$$ and $$z = y + y_1$$. Your new independent variable will be $$t$$ and your new dependent variable will be $$z$$; you'll see that this new differential equation in $$z$$ is homogeneous. Once you solve this equation, you need to plug back into get the solution in terms of $$y$$ and $$x$$.

Hope this helps. :)

 

[HallsofIvy]

Nice!

 

[em919]

When I do this the constants go away but what should I do with the other side of the d.eq?

 

[Metaleer]

Differentiate both sides in $$z = y + y_1$$; you'll see that $$z' = y'$$.