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`

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? y^(x-2x y^(prime))=y y y^(prime)=2x(y^(prime))^2+1 x y^(prime)=2y(y^(prime))^2+2 y^(y-4x y^(prime))=(y^(prime))^2

If x^2+y^2=1, then (a) y y^('')-2(y^(prime))^2+1=0 (b) yy^('')+(y^(prime))^2+1=0 (c) y y^('')+(y^(prime))^(-2)-1=0 (d) y y^('')+2(y^(prime))^2+1=0

If x^2+y^2=1,t h e n (a) y y^('')-2(y^(prime))^2+1=0 (b) yy^('')+(y^(prime))^2+1=0 (c) y y^('')+(y^(prime))^(-2)-1=0 (d) y y^('')+2(y^(prime))^2+1=0

If x^2+y^2=1,t h e n (a) y y^('')-2(y^(prime))^2+1=0 (b) yy^('')+(y^(prime))^2+1=0 (c) y y^('')+(y^(prime))^(-2)-1=0 (d) y y^('')+2(y^(prime))^2+1=0

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?

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

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

Solve each of the following initial value problem: y^(prime)+y=e^x ,\ y(0)=1/2

If x^2+y^2=1 , then a. y y''-2(y^(prime))^2+1=0 b. yy''+(y^(prime))^2+1=0 c. yy''+(y^(prime))^(-2)-1=0 d. yy''+2(y^(prime))^2+1=0

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)