TYPE – 1 VARIABLES SEPARABLE : If the differential equation can be expressed as ;
f (x) dx + g (y) dy = 0 then this is said to be variable – separable type
A general solution of this is given by ∫ f(x) dx + ∫ g(y) dy =c
where c is the arbitrary constant . Consider the example (dy/dx) = ex–y + x2 . e–y.
Examples :
(i) in(dy/dx)=ax+by Þ (dy/dx) =eax+by =eax .eby or ∫ e-by dy=∫ eax dx
(ii) x(y2+1) dx+y (x2 +1)dy=0
(iii) y’ sinx = y ln y ; |y|x=(p/2) =e
(iv) Find the foci of the conic passing through the point (1, 0) and satisfying the differential equaion (1 + y2) dx – xy dy = 0. Find also the equation of a circle touching the conic at (Ö2,1) and passing through one of its foci.
[Ans: (x-Ö2)2 + (y – 1)2 + l (Ö2 x –y-1) where l = -1/2 or 3 ; conic x2 – y2 = 1 (Rectangular Hyperbola)
(v) Find the nature of the curve for which the lengths of the
normal / Tangent at the point P is equal the radius vector
of the point P.
[Equation of normal : Y-y=-(1/m)(x,y)
ON2 = GN2
X2 + y2 = m2 y2 + y2
m=± x/y
with +ve x dx – y dy = 0
(vi) Find the curve for which the segment of the tangent contained between the co-ordinate axes is bisected by the point. Curve passes through (2, 3).
In the figure, find the curve if passes through (2, 0) and l(PA) = 2.
[ Ans: xy = 6] .
(vii) Find the curve y = f(x) (f(x) £0 and f(0) = 0) bounding a curvilineartrapezoid with the base [0, x] if
Area a (f (x))n+1 and f (1) = 1 (T/S) 0∫x y dx = k.(f(x))n+1 now differentiate both sides ; [Ans : f(x) = x1/n]
(viii) Show that the curve passing through (1, 2) for which the segment of the tangent between P and A is bisected at its point of intersectin with the y-axis is a parabla.
[Ans : y2 = cx Þ y2 = 4x]
Further show that if , P, Q, R are three points on this parabola such that three normals at them are concurrent on the line y = d then the sides of the DPQR touch another parabola
X2 = 2dy.
(ix) Find the curve for which area bounded by the curve the co-ordinate axes and a variable ordinate is equal to the length of the corresponding arc. Given that the curve passes through
(0, 1) (T/S-misc.)
\ 0∫x y dx = A ∫p dI =∫ Ö((dx)2 +(dy)2) =0∫x Ö[1+(dy/dx)2 dx]
No differentiable both sides. [Ans : & y ((ex +e-x)/2) & y= 1]
