Find the differential equation of all circles x² +y² + 2gx = 0 which pass through the origin and whose centres are on the X-axis.

Form the differential equation for \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \)=1 where a & b are arbitrary constants.

Form the differential equation for y = (A + Bx)e^3x where A and B are constants.

Solve: sec 2x dy - sin 5x sec² y dx = 0

Solve: cos²x dy + y.e^tanx dx = 0

Solve: (x²-yx²)dy + (y²+xy²)dx = 0

Solve: (x+y)²\(\frac { dy }{ dx } \) = 1

Find the equation of the curve passing through (1, 0) and which has slope 1+ \(\frac { y }{ x } \) at (x, y).

Show that the equation of the curve whose slope at any point is equal to y + 2x and which passes through the origin is y = 2(e^x-x-1).

Solve: (D²+1)y = 0 when x = 0, y = 2 and when x = \(\frac { \pi }{ 2 } \), y = -2.