The slope at any point of a curve y = f (x) is given by \(\frac{dy}{dx}=3x^2\) and it passes through (-1, 1). Then the equation of the curve is

If the solution of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\)represents a circle, then the value of a is

P is the amount of certain substance left in after time t. If the rate of evaporation of the substance is proportional to the amount remaining, then

The population P in any year t is such that the rate of increase in the population is proportional to the population. Then

Integrating factor of the differential equation \(\frac{d y}{d x}=\frac{x+y+1}{x+1}\) is 

The number of arbitrary constants in the particular solution of a differential equation of third order is

The number of arbitrary constants in the general solutions of order n and n +1 are respectively

If sin x is the integrating factor of the linear differential equation \(\frac { dy }{ dx } +Py=Q,\) then P is

The solution of the differential equation \(\frac{d y}{d x}=\frac{y}{x}+\frac{\phi\left(\frac{y}{x}\right)}{\phi^{\prime}\left(\frac{y}{x}\right)}\) is

The solution of \(\frac { dy }{ dx } ={ 2 }^{ y-x }\) is

The general solution of the differential equation \(\log \left(\frac{d y}{d x}\right)=x+y\) is

The solution of the differential equation \(\frac { dy }{ dx } =2xy\) is

The solution of the differential equation \(\frac { dy }{ dx } +\frac { 1 }{ \sqrt { 1-{ x }^{ 2 } } } =0\) is

If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\) When

The degree of the differential equation \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is

The integrating factor of the differential equation \(\frac{d y}{d x}+P(x) y=Q(x)\) is x, then P(x)

The integrating factor of the differential equation \(\frac { dy }{ dx } +y=\frac { 1+y }{ \lambda } \) is

The solution of \(\frac{d y}{d x}+p(x) y=0\) is

The solution of the differential equation \(2x\frac{dy}{dx}-y=3\) represents

The general solution of the differential equation \(\frac { dy }{ dx } =\frac { y }{ x } \) is

The differential equation of the family of curves y = Ae^x + Be^−x, where A and B are arbitrary constants is

The order of the differential equation of all circles with centre at (h, k ) and radius ‘a’ is

The order and degree of the differential equation \(\sqrt { sinx } (dx+dy)=\sqrt { cos x } (dx-dy)\) is

The differential equation representing the family of curves y = Acos(x + B), where A and B are parameters,is

The order and degree of the differential equation \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +{ \left( \frac { dy }{ dx } \right) }^{ 1/3 }+{ x }^{ 1/4 }=0\) are respectively