# Math Help - integration of (ax+by+c)/(ex+fy+g) = dy/dx, where a,c,c,e,f,g are constants

1. ## integration of (ax+by+c)/(ex+fy+g) = dy/dx, where a,c,c,e,f,g are constants

My maths book tells me that if we let X= x+k and y = Y+m, then dY/dX
equals (aX + bY)/(eX +fY) and then proceed as for a homogenous equation. My question is what happened to c and g and how is the new expression homogenous?

2. the values of k and m are taken such that c and g are removed

3. ok. Thanks. I still don't understand how removing a constant (c) can be the same as multiplying another constant (a or e) by X over the entire range in which x operates, even if another variable (X) is used. Could you help me, please? What ensures that the new expression tracks the original one?

4. to find the numbers k and m solve salmantaniously the equations

ax + by + c = 0 and ex + fy + g = 0

and substitute x = X - k and y = Y -m in given diff equation try this method with a question with numerical values

5. The substitution required depends on whether the two lines ax+by+c=0 and ex+fy+g=0 are intersecting lines, identical lines, or parallel, non-intersecting lines. If they are identical lines, then the DE simplifies greatly. If they are parallel, non-intersecting lines, then let u = ax+by+c, and rewrite the DE in terms of u and y. The resulting equation will be separable.

Lastly, if the coefficients, when equated to zero, represent intersecting lines, then find the point of intersection (p,q). Define new coordinates u = x-p, v = y-q, and rewrite the DE in terms of these coordinates. The resulting DE will be homogeneous.