Question

1) Find d^2y / dx^2 given that y^3 - x^2 = 4


2)If y = xª + ( 1 / xª ) , prove that

x^2 . y'' + xy' – a^2 . y = 0


3) Find dy/dx for the following function.
cos ( x - y ) = ( y ) ( 2x + 1 ) ^3

4)Find the values of A and B that make f(x) differentiable at
x = 0

f(x) = A + 2x + 3, if x > 0
AND
f(x) = 5 + B (sin²x / |x|) , if x < 0

Answer 1

y^3 - x^2 = 4
3y^2 y' - 2x = 0
y' = 2x/3y^2

6y y'^2 + 3y^2 y'' - 2 = 0

y'' = (2 - 6yy'^2)/3
= (2 - 6y(2x/3y^2)^2)/3
_______________

y = x^a + 1/x^a
y' = ax^(a-1) - a/x^(a+1)
y'' = a(a-1)x^(a-2) + a(a+1)/x^(a+2)

x^2 y'' = a(a-1)x^a + a(a+1)/x^a
xy' = ax^a - a/x^a
-a^2 y = -a^2 x^a - a^2/x^a

Add 'em up: 0
_______________

cos(x-y) = y(2x+1)^3
-sin(x-y)(1-y') = y'(2x+1)^3 + 6y(2x+1)^2
y'(sin(x-y) - y'(2x+1)^3 = xsin(x-y) + 6y(2x+1)^2
carry on
__________________

lim sin^2 x/x = 2sincos/1 = 0

f(0) = A + 3
f(0) = 5 + 0B = 5

So, if A=2, then the limit exists from both sides

Answer 2

thank u so much for ur effort and time
helped me alot!!!!!