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 .