3. Differential Equation Reducible to Variables Separable Method
A differential equation of the form dy / dx = f( ax + by + c) can be reduced to variables separable form by substituting
ax + by + c = z => a + b dy / dx = dz / dx
The given equation becomes 1 / b( dz / dx – a) f( z) => dz / dx = a + b f( z) => dz / a + bf( z) = dx
Hence, the variables are separated in terms of z and x.
4. Homogeneous Differential Equation
A function f( x, y) is said to be homogeneous of degree n, if f( λx, λy) = λ n f( x, y)
Suppose a differential equation can be expressed in the form dy / dx = f( x, y) / g( x, y) = F( y / x)
where, f( x, y) and g( x, y) are homogeneous function of same degree. To solve such types of equations, we put y = vx
=> dy / dx = v + x dv / dx. The given equation, reduces