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`

For each of the following initial value problems verify that the accompanying functions is a solution. (i) x(dy)/(dx)=1,\ y(1)=0 => y=logx (ii) (dy)/(dx)=y ,\ y(0)=1 => y=e^x (iii) (d^2y)/(dx^2)+y=0,\ y(0)=0,\ y^(prime)(0)=1 => y=sinx (iv) (d^2y)/(dx^2)-(dy)/(dx)=0,\ y(0)=2,\ y^(prime)(0)=1 => y=e^x+1

Solve the following differential equation for a particular solution: y-x(dy)/(dx) = x+y (dy)/(dx) , where y=0 and x=1.

Solve the following differential equation: (1-x^2)dy+x y\ dx=x y^2dx

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

Solve the following differential equation: y^2dx+(x^2-x y+y^2)dy=0

Solve the following differential equation: (1-x^2)dy+x y\ dx\ =x y^2 dx

Solve the following differential equation : y^2dx+(x^2-x y+y^2)dy=0

Solve the following differential equation: \ (y^2-2x y)dx=(x^2-2x y)dy

Solve the following differential equations: (x^2+3x y+y^2)dx-x^2dy=0

Which of the following differential equations has y = x as one of its particular solution? (A) (d^2y)/(dx^2)-x^2(dy)/(dx)+x y=x (B) (d^2y)/(dx^2)+x(dy)/(dx)+x y=x (C) (d^2y)/(dx^2)-x^2(dy)/(dx)+x y=0 (D) (d^2y)/(dx^2)+x(dy)/(dx)+x y=0