For each of the following differential equations, determine its order, degree (if exists)
\({ x }^{ 2 }\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +{ \left[ 1+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }=0\)

For each of the following differential equations, determine its order, degree (if exists)
\(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } +5\frac { dy }{ dx } +\int { ydx } ={ x }^{ 3 }\)

Find the differential equation of the family of all nonhorizontal lines in a plane.

Form the differential equation of all straight lines touching the circle x² + y² = r².

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

If F is the constant force generated by the motor of an automobile of mass M, its velocity is given by M \(\frac{dV}{dt}\)= F-kV, where k is a constant. Express V in terms of t given that V = 0 when t = 0.

The velocity v , of a parachute falling vertically satisfies the equation \(\\ \\ \\ \\ \\ \\ \\ v\frac { dv }{ dx } =g\left( 1-\frac { { v }^{ 2 } }{ { k }^{ 2 } } \right) \\ \\ \), where g and k are constants. If v and x are both initially zero, find v in terms of x.

Solve the following differential equations:
ydx + (1 +x²) tan^-1 xdy = 0

Solve the following differential equations
\(\left[ x+y\quad cos\left( \frac { y }{ x } \right) \right] dx=x\ cos\left( \frac { y }{ x } \right) dy\)

Solve the following differential equations
\(x\frac { dy }{ dx } =y-x{ cos }^{ 2 }\left( \frac { y }{ x } \right) \)

Solve the Linear differential equation:
\(x\frac { dy }{ dx } +y=xlogx\)

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.

A pot of boiling water at 100^o C is removed from a stove at time t = 0 and left to cool in the kitchen. After 5 minutes, the water temperature has decreased to 80^o C , and another 5 minutes later it has dropped to 65^oC. Determine the temperature of the kitchen.

A tank initially contains 50 litres of pure water. Starting at time t = 0 a brine containing with 2 grams of dissolved salt per litre flows into the tank at the rate of 3 litres per minute. The mixture is kept uniform by stirring and the well-stirred mixture simultaneously flows out of the tank at the same rate. Find the amount of salt present in the tank at any time t > 0.