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

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

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

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

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

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

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

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

The order and degree of the differential equation \({ \left[ \left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +\left( \frac { dy }{ dx } \right) \right] }^{ \frac { 1 }{ 2 } }=\frac { { d }^{ 3 }y }{ { dx }^{ 3 } } \) are __________

The solution of \(\frac{dy}{dx}+y\) cot x = sin 2x is ___________

The solution of log \(\left( \frac { dy }{ dx } \right) \) = ax + by is______.

The differential equation of x²y = k is _________.

The I.F. of (1+y²) dx = (tan^-1-t-x) dy is ________.

For each of the following differential equations, determine its order, degree (if exists)
\(\frac { dy }{ dx } +xy=cotx\)

For each of the following differential equations, determine its order, degree (if exists)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)

Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop.

Find the differential equation of the family of all non-vertical lines in a plane.

Show that y = e^−x + mx + n is a solution of the differential equation e^x \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

Determine the order and degree (if exists) of the following differential equations: 
\({ \left( \frac { { d }^{ 4 }y }{ { dx }^{ 4 } } \right) }^{ 3 }+4{ \left( \frac { dy }{ dx } \right) }^{ 7 }+6y=5cos3x\)

Find the differential equation of the family of parabolas y² = 4ax, where a is an arbitrary constant.

Solve: \(\frac{dy}{dx}+y=e^{-x}\)

Determine the order and degree of \(\frac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

Form the Differential Equation representing the family of curves y = A cos(x + B) where A and B are parameters.

Find the differential equation of the family of circles passing through the origin and having their centres on the x -axis.

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

Find the particular solution of (1+ x³)dy − x² ydx = 0 satisfying the condition y(1) = 2.

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.

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

Form the differential equation for y = e^-2x [A cos 3x-B sin 3x]

Solve: x\(\frac{dy}{dx}\)+ 2y = x²

Verify that y=-x-1 is a solution of the D.E (y-x)dy-(y²-x²)dx=0

Solve:\(\frac { dy }{ dx } =\frac { 1-cosx }{ 1+cosx } \)

Solve : ydx+(x-y²)dy=0

Find the equation of the curve whose slope is \(\frac { y-1 }{ { x }^{ 2 }+x } \) and which passes through the point (1, 0).

Solve (x² -3y²) dx + 2xydy = 0.

Solve the Linear differential equation \((1+x+{ xy }^{ 2 })\frac { dy }{ dx } +(y+{ y }^{ 3 })=0\)

Solve : (1+y²)(1 + log x)dx + x dy = 0, given that x = 1,y = 1.

Solve : \({ 2x }^{ 2 }\left( \frac { dy }{ dx } \right) -2xy+{ y }^{ 2 }=0,y(e)=e\)

Solve : \(\frac { dy }{ dx } =-\frac { x+ycos }{ 1+sinx } \) .Also find the domain of the function.

Water at temperature 100℃ cools in 5 minutes to 80℃ in a room of temperature 30℃. Find (i) the temperature of water after 10 minutes. (ii) the time when the temperature is 40℃.