If `x=varphi(t), y=psi(t),t h e n(d^(2y))/(dx^2)` is (a)`(varphi^(prime)psi^('')-psi'varphi' ')/((varphi^(prime))^2)` (b) `(varphi^(prime)psi^('')-psi'varphi' ')/((varphi^(prime))^3)` (c)`varphi^('')/psi^('')` (d) `psi^('')/varphi^('')`

If x=t^2,y=t^3,t h e n(d^2y)/(dx^2)= (a) 3/2 (b) 3/((4t)) (c) 3/(2(t)) (d) (3t)/2

If varphi(x) is differentiable function AAx in R and a in R^+ such that varphi(0)=varphi(2a),varphi(a)=varphi(3a)a n dvarphi(0)!=varphi(a) then show that there is at least one root of equation varphi^(prime)(x+a)=varphi^(prime)(x)in(0,2a)

The general solution of the differential equation, y^(prime)+yvarphi^(prime)(x)-varphi^(prime)(x) varphi(x)=0 , where varphi(x) is a known function, is

Statement 1: int({f(x)varphi^(prime)(x)-f^(prime)(x)varphi(x)})/(f(x)varphi(x)) -logf(x)dx=1/2{(varphi(x))/(f(x))}^2+c Statement 2 : int(h(x))^n h^(prime)(x)dx=((h(x))^(n+1))/(n+1)+c

Let inte^x{f(x)-f^(prime)(x)}dx=varphi(x)dot Then inte^xf(x)dx is varphi(x)= e^xf(x)

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)

If y=a x^(n+1)+b x^(-n),t h e nx^2(d^2y)/(dx^2) is equal to n(n-1)y (b) n(n+1)y (c) n y (d) n^2y

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx

If varphi(x) is a polynomial function and varphi^(prime)(x)>varphi(x)AAxgeq1a n dvarphi(1)=0, then varphi(x)geq0AAxgeq1 varphi(x)

Suppose fa n dg are functions having second derivative f'' and g' ' everywhere. If f(x)dotg(x)=1 for all xa n df^(prime)a n dg' are never zero, then (f^('')(x))/(f^(prime)(x))-(g^('')(x))/(g^(prime)(x))e q u a l (a) (-2f^(prime)(x))/f (b) (2g^(prime)(x))/(g(x)) (c) (-f^(prime)(x))/(f(x)) (d) (2f^(prime)(x))/(f(x))