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Practice each question, then open the answer or worked solution.
Find the equation of a curve passing through origin if the slope of the tangent to the curve at any point (x, y) is equal to the square of the difference of the abscissa and ordinate of the point.
We know that,
The slope of the tangent of the curve = dy/dx
And the difference between the abscissa and ordinate = x – y
So, as given in the question we have
dy/dx = (x – y)2
Taking, x – y = v
Find the equation of a curve passing through the point (1, 1). If the tangent drawn at any point P (x, y) on the curve meets the co-ordinate axes at A and B such that P is the mid-point of AB.
Let’s take P(x , y) be any point on the curve and AB be the tangent to the given curve at P.
Also given, P is the mid-point of AB
So, the coordinates of A and B are (2x, 0) and (0, 2y) respectively.
The slope of the tangent AB = (2y – 0)/ (0 – 2x) = -y/x
Thus, the required equation is xy = 1
Solve :
Objective Type
Choose the correct answer from the given four options in each of the Exercises from 34 to 75 (M.C.Q)
The degree of the differential equation
(A) 1 (B) 2 (C) 3 (D) not defined
Correct option is (D) not defined.
Since the value of sin (dy/dx) on expansion will be in increasing power of dy/dx, the degree of the given differential equation is not defined.
The degree of the differential equation
(A) 4 (B) 3/2 (C) not defined (D) 2
Correct option is (D) 2.
Given differential equation is
Thus, the degree of the given differential equation is 2.
The order and degree of the differential equation respectively, are
(A) 2 and not defined (B) 2 and 2 (C) 2 and 3 (D) 3 and 3
Correct option is (A) 2 and not defined.
Given differential equation is
As the degree of dy/dx is in fraction its undefined and the degree is 2.
If y = e–x (A cos x + B sin x), then y is a solution of
Correct option is (C).
Given equation, y = e–x (A cos x + B sin x)
Differentiating on both the sides, w.r.t. x, we get
The differential equation for y = A cos αx + B sin αx, where A and B are arbitrary constants is
Correct option is (B).
Given equation is y = A cos ax + B sin ax
Differentiating both sides w.r.t. x, we have
