Homogeneous Differential Equation Solver
Solve first-order homogeneous differential equations using variable substitution (y=vx). Detects degree of homogeneity and outlines separation of variables.
Homogeneous ODE Check: dy/dx = Ax + ByCx + Dy
About
A first-order differential equation is homogeneous if it can be written in the form dy/dx = F(y/x). Identifying this structure is the key to solving it. The standard method involves the substitution y = vx, which transforms the complex ODE into a separable equation involving v and x.
This tool helps students and engineers verify if an equation fits the homogeneous pattern and demonstrates the initial substitution steps required to reach the general solution.
Formulas
For a function M(x,y) to be homogeneous of degree n:
M(tx, ty) = tn M(x, y)
If the ODE is Mdx + Ndy = 0 and both M and N are homogeneous of the same degree, we use y = vx.
Reference Data
| Form | Substitution | Resulting Separable Form |
|---|---|---|
| y' = f(y/x) | y = vx | dvf(v) − v = dxx |
| xdy − ydx = 0 | y = vx | dv/v = 0 (Trivial) |