If a function is represented parametrically be the equations `x=(1+(log)_e t)/(t^2); y=(3+2(log)_e t)/t ,` then which of the following statements are true? (a)`y^('')(x-2x y^(prime))=y` (b)`y y^(prime)=2x(y^(prime))^2+1` (c)`x y^(prime)=2y(y^(prime))^2+2` (d)`y^('')(y-4x y^(prime))=(y^(prime))^2`

Consider the family of all circles whose centers lie on the straight line y=x . If this family of circles is represented by the differential equation P y^(primeprime)+Q y^(prime)+1=0, where P ,Q are functions of x , y and y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)), then which of the following statements is (are) true? (a) P=y+x (b) P=y-x (c) P+Q=1-x+y+y^(prime)+(y^(prime))^2 (d) P-Q=x+y-y^(prime)-(y^(prime))^2

y=2^(1/((log)_x4) , then find x in terms of y.

If a curve is represented parametrically by the equation x=f(t) and y=g(t)" then prove that "(d^(2)y)/(dx^(2))=-[(g'(t))/(f'(t))]^(3)((d^(2)x)/(dy^(2)))

The solution of the differential equation y^(prime)y^(primeprimeprime)=3(y^(primeprime))^2 is

The condition that one of the straight lines given by the equation a x^2+2h x y+b y^2=0 may coincide with one of those given by the equation a^(prime)x^2+2h^(prime)x y+b^(prime)y^2=0 is (a b^(prime)-a^(prime)b)^2=4(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (a b^(prime)-a^(prime)b)^2=(h a^(prime)-h^(prime)a)(b h^(prime)-b^(prime)h) (h a^(prime)-h^(prime)a)^2=4(a b^(prime)-a^(prime)b)(b h^(prime)-b^(prime)h) (b h^(prime)-b^(prime)h)^2=4(a b^(prime)-a^(prime)b)(h a^(prime)-h^(prime)a)

The solution of the differential equation y^(prime prime prime)-8y^(prime prime)=0, where y(0)=1/8,y^(prime)(0)=0,y^(prime prime)(0)=1 , is

If the circle x^2+y^2+2gx+2fy+c=0 bisects the circumference of the circle x^2+y^2+2g^(prime)x+2f^(prime)y+c^(prime)=0 then prove that 2g^(prime)(g-g^(prime))+2f^(prime)(f-f^(prime))=c-c '

Let y=f(x) be a parabola, having its axis parallel to the y-axis, which is touched by the line y=x at x=1. Then, (a) 2f(0)=1-f^(prime)(0) (b) f(0)+f^(prime)(0)+f^(0)=1 (c) f^(prime)(1)=1 (d) f^(prime)(0)=f^(prime)(1)

Let y(x) be a solution of the differential equation (1+e^x)y^(prime)+y e^x=1. If y(0)=2 , then which of the following statements is (are) true? (a) y(-4)=0 (b) y(-2)=0 (c) y(x) has a critical point in the interval (-1,0) (d) y(x) has no critical point in the interval (-1,0)

A function y=f(x) satisfies the differential equation (d y)/(d x)+x^2 y=-2 x, f(1)=1 . The value of |f^( prime prime)(1)| is