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'.