Similarly, for the second differential equation dx / dy + Rx = S, the integrating factor, IF = e ∫ R dy and the general solution is
x( IF) = ∫ S( IF) dy + C
8. Differential Equation Reducible to Linear Form
Bernoulli’ s Equation An equation of the form dy / dx + Py = Qy n, where P and Q are functions of x along or constants, is called
Bernoulli’ s equation.
Divide both the sides by y n, we get y-n dy / dx + Py-n + 1 = Q
Put y-n + 1 = z
=>(-n + 1) y-n dy / dx = dz / dx The equation reduces to
1 / 1 – n dz / dx + Pz = Q => dz / dx +( 1 – n) Pz = Q( 1 – n) which is linear in z and can be solved in the usual manner.
9. Clairaut Form for Differential Equation
Differential equation y = Px + f( p), where P = dy / dx …( i)
is called clairaut form of differential equation. In which, get its general solution by replacing P from C.