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

The slope of the tangent to the curve at any point is the reciprocal of four times the ordinate at that point. The curve passes through (2, 5). Find the equation of the curve.

Form the differential equation by eliminating the arbitrary constants A and B from y = A cos x + B sin x.

Solve : \(\frac { dy }{ dx } =\sqrt { 4x+2y-1 } \)

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

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

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

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

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).