Find the differential equation of the family of parabolas with vertex at (0, −1) and having axis along the y-axis.

Find the differential equation corresponding to the family of curves represented by the equation y = Ae^8x + Be^-8x, where A and B are arbitrary constants.

Find the differential equation of the curve represented by xy = ae^x + be^−x + x².

Show that y = ae^-3x + b, where a and b are arbitary constants, is a solution of the differential equation\(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3\frac { dy }{ dx } =0\)

Find the differential equation of the family of all ellipses having foci on the x -axis and centre at the origin.

Solve the following differential equations:
\(sin\frac { dy }{ dx } =a,y(0)=1\)

Solve the following differential equations:
(e^y+1) cos x dx + e^y sin x dy = 0

Solve the following differential equations:
tan y\(\frac{dy}{dx}\) = cos(x+y)+cos(x-y)

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +2y-x^2logx=0\)

The engine of a motor boat moving at 10 m/s is shut off. Given that the retardation at any subsequent time (after shutting off the engine) equal to the velocity at that time. Find the velocity after 2 seconds of switching off the engine.