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