TYPE -8 EQUATIONS REDUCIBLE TO LINEAR FORM : (BERNOULLI'S EQUATION)
The equation where (dy/dx)+py =Q .yn where P & Q functions of x, is reducible to the linear form by dividing it by yn & then substituting y–n+1 = Z. Its solution can be obtained as in the normal case
Examples:
(i)
(ii) Find the curve such that the y-intercept cut off by the tangent on any arbitrary point is
(a) proportional to the square of the ordinate of the the point of tangency.
(b) proportional to the cube of the ordinate of the point of tangency.
General :
(i) In an equation of the form : y f (xy) dx + xg (xy) dy = 0 the variables can be separated by the substitution xy = v. Consider the example
[Solution : (v2 + v2 + v +1) = –(xdy/ydx) (v3 – v2 – v +1) .............(1)
where x (dy/dx) + y = (dv/dx) Þ x (dy/dx) =(dv/dx) –y .............(2)
from (1) and using (2)
xy -(1/xy ) -2 Iny =c Answer]
(ii)
(iii) Series of substitution (xcosy – y sin y) dy + (x sin y + y cosy ) dx = 0
start x sin y + y cos y = t
(iv)
