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

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

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

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

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)
\({ \left( \frac { d^2y }{ dx^2 } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)

Express each of the following physical statements in the form of differential equation.
A saving amount pays 8% interest per year, compounded continuously. In addition, the income from another investment is credited to the amount continuously at the rate of Rs. 400 per year.

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

Determine the order and degree (if exists) of the following differential equations: 
\(3\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ={ \left[ 4+{ \left( \frac { dy }{ dx } \right) }^{ 2 } \right] }^{ \frac { 3 }{ 2 } }\)

Show that y = 2(x²−1)+Ce^−x2 is a solution of the differential equation \(\frac { dy }{ dx } +2xy-4{ x }^{ 3 }=0\)

The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple?

A radioactive isotope has an initial mass 200mg, which two years later is 50mg. Find the expression for the amount of the isotope remaining at any time. What is its half-life? (half-life means the time taken for the radioactivity of a specified isotope to fall to half its original value).

Solve \(\frac { dy }{ dx } =\frac { x-y+5 }{ 2(x-y)+7 } .\)

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