`(x^2+1)(dy)/(dx)+2x y=sqrt(x^2+4)`

x(dy)/(dx)-y=2sqrt(y^(2)-x^(2))

If y= sqrt ( x) + (1)/( sqrtx ) , prove that 2x (dy)/( dx ) + y=2 sqrt (x ) .

If y = sqrt(x) + (1)/(sqrt(x)) prove that 2x(dy)/(dx) + y = 2sqrt(x)

If y = sin^(-1) x^2 sqrt(1 - x^2) + xsqrt(1 - x^4) , show that (dy)/(dx) - (2x)/(sqrt(1 - x^4)) = 1/(sqrt(1 - x^2))

For each of the following differential equations verify that the accompanying functions a solution. (i) x(dy)/(dx)=y => y=a x (ii) x+y(dy)/(dx)=0 => y=+-sqrt(a^2-x^2) (iii) x(dy)/(dx)y+y^2 => y=a/(x+a) (iv) x^3(d^2y)/(dx^2)=1 => y=a x+b+1/(2x) (v) y=((dy)/(dx))^2 => y=1/4(x+-a)^2

(y)/(x)(dy)/(dx)=sqrt(1+x^(2)+y^(2)+x^(2)y^(2))

Solve the following differential equations. (i) (dy)/(dx) =(1+y^(2))/(1+x^(2)) (ii) (dy)/(dx) = (sqrt(1-y^(2)))/(sqrt(1-x^(2))) (iii) (dy)/(dx) = 2y tan hx (iv) sqrt(1+x^(2))dx + sqrt(1+y^(2))dy = 0 (v) (dy)/(dy) = e^(x-y)+x^(2)e^(-y)

If y = sqrt(1+sqrt(1+x^(4))) , prove that y * (y^(2) - 1)(dy)/(dx) = x^(3) .

If y=(x^(2))/(2)+(x)/(2) sqrt(x^(2)+1)+log sqrt(x+sqrt(x^(2)+1)) , prove that, 2y=x (dy)/(dx)+log((dy)/(dx))

If y=sqrt(x)+(1)/(x), Show that 2x(dy)/(dx)+y=2sqrt(x)